Secondary II
Algebraic Operations
Teacher Edition
Unit 4
Northern Utah Curriculum Consortium
Project Leader
Sheri Heiter
Weber School District
Project Contributors
Ashley Martin
Bonita Richins
Craig Ashton
Davis School District
Cache School District
Cache School District
Gerald Jackman
Jeff Rawlins
Jeremy Young
Box Elder School District
Box Elder School District
Box Elder School District
Kip Motta
Marie Fitzgerald
Mike Hansen
Rich School District
Cache School District
Cache School District
Robert Hoggan
Sheena Knight
Teresa Billings
Cache School District
Weber School District
Weber School District
Wendy Barney
Helen Heiner
Susan Summerkorn
Weber School District
Davis School District
Davis School District
Lead Editor
Allen Jacobson
Davis School District
Technical Writer/Editor
Dianne Cummins
Davis School District
NUCC | Secondary II Math i
Table of Contents
4.1
STRUCTURE OF EXPRESSIONS .........................................................................................................1
Teacher Notes ..................................................................................................................................................1
Triple Match.................................................................................................................................................1
Assignment: Ready, Set ...............................................................................................................................3
Classroom Activity: Go! ..............................................................................................................................4
Expressions 1 Bell Quiz and Answers .........................................................................................................4
Mathematics Content .......................................................................................................................................5
Triple Match.....................................................................................................................................................6
Ready, Set ........................................................................................................................................................7
Go! ...................................................................................................................................................................9
Expressions 1 Bell Quiz .................................................................................................................................11
4.2
OPERATIONS WITH POLYNOMIALS ..............................................................................................12
Teacher Notes ................................................................................................................................................12
Building Polynomial Functions Activity ...................................................................................................12
Assignment Ready, Set (Operations with Polynomials Worksheet) ..........................................................14
Assignment Go! (Use either option) ..........................................................................................................15
Option 1: Polynomial Puzzles Task ...........................................................................................................15
Option 2: Pascal’s Triangle Task ...............................................................................................................16
Polynomial Bell Quiz 1 ..............................................................................................................................16
Polynomial Bell Quiz 2 ..............................................................................................................................17
Polynomial Bell Quiz 3 ..............................................................................................................................17
Mathematics Content .....................................................................................................................................18
Building Polynomial Functions .....................................................................................................................19
Operations with Polynomials Worksheet .......................................................................................................22
Polynomial Puzzler ........................................................................................................................................24
Pascal’s Triangle Task ...................................................................................................................................25
Polynomial 1 Bell Quiz..................................................................................................................................27
Polynomial 2 Bell Quiz..................................................................................................................................28
Polynomial 3 Bell Quiz..................................................................................................................................29
4.3
FACTORING POLYNOMIALS ...........................................................................................................30
Teacher Notes ................................................................................................................................................30
Option 1: Polynomial Factoring Practice Worksheet.................................................................................32
Option 2: Polynomial Puzzle .....................................................................................................................33
Factoring Bell Quiz 1 .................................................................................................................................34
Mathematics Content .....................................................................................................................................35
NUCC | Secondary II Math ii
Polynomial Factoring Practice .......................................................................................................................36
4.4
FACTORING SPECIAL PRODUCTS ..................................................................................................37
Teacher Notes ................................................................................................................................................37
Option 1. Special Products Factoring Worksheet ......................................................................................41
Option 2. Special Products and Factoring Task .........................................................................................42
Option 3. I Have, Who Has Factoring Worksheet .....................................................................................43
Special Products Bell Quiz 1 .....................................................................................................................44
Mathematics Content .....................................................................................................................................45
Patterns with Polynomial Products ................................................................................................................46
Special Products Factoring Worksheet ..........................................................................................................47
Special Products and Factoring Task .............................................................................................................49
I Have, Who Has Factoring Worksheet .........................................................................................................55
Special Products Bell Quiz ............................................................................................................................58
4.5
INTRO TO COMPLEX NUMBERS .....................................................................................................59
Teacher Notes ................................................................................................................................................59
John and Betty’s Story ...............................................................................................................................59
Ready, Set, Go! ..........................................................................................................................................61
Complex Numbers Worksheet ...................................................................................................................61
Mathematics Content .....................................................................................................................................62
Complex Numbers: ........................................................................................................................................63
The Story of John and Betty Guided Notes ...................................................................................................64
4.6
OPERATIONS WITH COMPLEX NUMBERS ...................................................................................65
Teacher Notes ................................................................................................................................................65
Mix –N –Match ..........................................................................................................................................65
Assign Ready, Set, Go! Worksheet ............................................................................................................67
Mathematics Content .....................................................................................................................................68
Teacher’s Master of Mix –N –Match.............................................................................................................69
Student copy of Mix –N –Match....................................................................................................................70
Ready, Set, Go! ..............................................................................................................................................71
H4.7 FACTORING WITH COMPLEX NUMBERS .....................................................................................72
Teacher Notes ................................................................................................................................................72
Mathematics Content .....................................................................................................................................74
Ready, Set, Go! ..............................................................................................................................................75
Practice Exam Secondary II Unit 4 ....................................................................................................................76
Exam Secondary II Unit 4..................................................................................................................................78
NUCC | Secondary II Math iii
Unit 4.1
4.1 STRUCTURE OF EXPRESSIONS
Teacher Notes
Time Frame: One 40 minute class period
Materials Needed: Scissors
Note: This will be a review lesson for most students, except possibly the degree of a
polynomial and the types of polynomials.
Launch
Triple Match Print one copy for each pair of students.
Instructions: Students will work in pairs and cut out the terms and match the
vocabulary word with the definition and an example. Select students to present their
answers to the class. (Students should have learned these terms previously.)
Explore
Dealing with Expressions
 expression = mathematical phrase that contains operations, numbers and/or variables. (no =)
Examples of expressions: 4, 4 + d, 𝟓𝒎𝟐 − 𝟐𝒎, k
 Write an algebraic expression for “6 apples and 2 mangos”. 6a + 2m

Write an English statement for
2+𝑚2
3
. The sum of 2 and m squared divided by 3.
NUCC | Secondary II Math 1
Unit 4.1

Use Algebra Tiles to model the expression: 3𝑥 2 + 𝑥 − 2 . Draw your model.
Algebra tiles help connect what is happening with polynomials visually. They also connect geometry
and algebra using area. This website has an excellent PowerPoint on the use of algebra tiles.
http://mathbits.com/mathbits/AlgebraTiles/AlgebraTiles.htm


*
monomial = a number or variable or the product of numbers and variables. Also called a term.
Examples of monomials: 4, k, 𝟓𝒎𝟐
polynomial = a monomial or the sum or difference of monomials. Examples of polynomials: 4, 4
+ d, 𝟓𝒎𝟐 − 𝟐𝒎, k
compare and contrast expressions and polynomials
Discuss
Polynomial Anatomy 101
𝑬𝒙: 𝟑𝒙𝟐 − 𝒙 + 𝟕 .
Parts of a polynomial






term = parts of the polynomial that are added or subtracted. (terms in the example are 𝟑𝒙𝟐 ,
−𝒙, 𝟕)
variable = unknown quantity represented by a symbol, usually a letter. (variable in the example
is x)
coefficient = a number multiplied by a variable. (coefficients in the example are 3 and –1)
operation = adding, subtracting, multiplying or dividing. (Operations in the example are
multiplying, subtracting and adding)
constant = unchanging, a term with only a number. (constant in the example is 7)
degree = the highest degree of the terms of the polynomial. (degree in the example is 2)
State the degree of each polynomial.
*
Find the degree of each term and the highest degree is the degree of the polynomial.
Ex. 4𝑥 + 2𝑥 3 − 𝑥 2 + 11
1
3
2
0
Ex. 3𝑎𝑏 4 + 4𝑎5 𝑏 2 − 𝑎7 𝑏
5
3
7
8
8
Write the polynomial 𝑥 4 𝑦 + 2𝑥 5 − 3𝑥 + 9𝑥 3 𝑦 6 in standard form in regards to x.
𝟐𝒙𝟓 + 𝒙𝟒 𝒚 + 𝟗𝒙𝟑 𝒚𝟔 − 𝟑𝒙
NUCC | Secondary II Math 2
Unit 4.1
Types of polynomials (named by the degree of the polynomial)



linear = degree of 1 (Ex: 4x + 7)
quadratic = degree of 2 (Ex: 𝟐𝒙𝟐 + 𝟓𝒙 − 𝟔)
cubic = degree of 3 (Ex: 𝟐𝒙𝟑 + 𝟐𝒙𝟐 − 𝟓𝒙 − 𝟔)
*
(leave the degree of 4 and 5 for the student to research)
Assignment: Ready, Set Reviewing expressions
NUCC | Secondary II Math 3
Unit 4.1
Classroom Activity: Go! Group Task
Expressions 1 Bell Quiz and Answers
NUCC | Secondary II Math 4
Unit 4.1
Mathematics Content
Cluster Title: Interpret the structure of expressions.
Standard A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
(For example, interpret P(1 + 4)n as the product of P and a factor not depending on P.)
Concepts and Skills to Master



Identify the parts of an expression, such as terms, factors, and coefficients, bases,
exponents, and constant.
Explain the meaning of the part in relationship to the entire expression and to the context
of the problem.
Understand that the product of two binomials is the sum of monomial terms. For example
the product of (3x + 2) and (x – 5) is the sum of 3x2, -13x, and -10.
Critical Background Knowledge


Understand the meaning of symbols indicating mathematical operations, implied
operations, the meaning of exponents, and grouping symbols.
Understand the meaning of a rational exponent (Secondary II: N.RN.2).
Academic Vocabulary
Factors, coefficients, terms, exponent, base, constant, variable, binomial, monomial, polynomial
Suggested Instructional Strategies


Connect to quadratic functions and
transformations. Identify the role of the
part in possible transformations.
Example: What role do j and k play in
(x – h)2 + k ?
Connect to area models.
Skills:


Given the quadratic 25x^2 + 30x + 9,
justify that it is a perfect square trinomial.
Use what you know about square roots
to rewrite x2 – 6 as a difference of two
squares.
Some Useful Websites:
http://mathbits.com/mathbits/AlgebraTiles/AlgebraTiles.htm
NUCC | Secondary II Math 5
Unit 4.1
Triple Match
NUCC | Secondary II Math 6
Unit 4.1
Name ________________________________________________
Period _________
Date _____________________
Ready, Set
Ready
Write an algebraic expression for each statement.
1.
Fifteen apples plus 4 bananas.
2.
Five ipods minus a computer.
Write an English statement for each algebraic expression.
3.
4.
(x + 7)(x – 4)
5𝑚2
6
+𝑚
Use Algebra Tiles to model the expressions. Draw your model.
5.
5𝑥 2 + 𝑥 + 2
6.
𝑥 2 − 4𝑥 + 3
7.
3𝑥 − 6
8.
3𝑥 − 2𝑥 2
Set
State the degree of each polynomial.
9. 2𝑥 + 𝑥 3 − 5𝑥 2 + 8
10.
5𝑎𝑏 3 + 4𝑎7 − 𝑎2 𝑏
11.
𝑥 3 𝑦 5 + 3𝑥 4 𝑦 3 − 𝑥 5
12.
2𝑥 2 + 4𝑥 − 𝑥 5 − 1
13.
2𝑥 + 2
14.
5
NUCC | Secondary II Math 7
Unit 4.1
If possible, determine 2 polynomials with a sum and product that have the following degrees.
15.
The sum has a degree of 2 and the product has a degree of 4.
16.
The sum has a degree of 3 and the product has a degree of 5.
17.
The sum has a degree of 2 and the product has a degree of 6.
18.
The sum and the product have the same degree.
19.
Complete the table.
DEGREE
POLYNOMIAL
NAME
EXAMPLE
BASIC GRAPH SHAPE
1
2
3
4
5
State each polynomial in standard form in regard to x.
20.
x 8 + 2x 4 − 6x 7 + x
21.
x 3 yz + yz + x 2 y 3 z − x
22.
4 − 2x 2 + 3x 4 + x 3
23.
5x 3 y + 6x 5 − 4x + x 7 y 2
24.
xy 5 + 6x 2 − x 3 y
25.
2x − 𝑥 2 + 𝑥 3 + 3
NUCC | Secondary II Math 8
Unit 4.1
Go!
Names in group ________________________________________________ Period ___________
Date _____________________
1. Dinner is served every night at St. Ann’s shelter. Write an expression for each of the supplies St.
Ann’s may need to serve dinner.
a. For each person, St. Ann’s needs ¼ cup of uncooked rice.
b. For every 10 adults, St. Ann’s needs 0.8 loaves of bread.
c. St. Ann’s needs half a pound of meat for every 3 people.
d. For each person, 2.3 cups of beverages are needed.
e. 9 more sets of dishes are needed than the number of people.
f. For every 25 people, St. Ann’s needs 2 servers.
Complete the table so that it may be used as a quick reference for St. Ann’s shelter.
EXPRESSION
50
PEOPLE
100
PEOPLE
250
PEOPLE
RICE
BREAD
MEAT
BEVERAGES
DISHES
SERVERS
2. Why are expressions useful?
3. Why are variables useful?
NUCC | Secondary II Math 9
Unit 4.1
Expressions and Number Tricks
4. Choose a number. Add 4. Subtract 2. Multiply by 2. Subtract 4.
Write your number and your solution.
a. Choose 3 more numbers and follow the same processes as you did in problem 4. Write your
numbers and their solutions.
b. What did you notice?
c. Write an expression for this math problem.
5. As a group make up a number trick with at least 4 steps, you will be presenting your trick to the
class and we will try to figure out how it works.
.
NUCC | Secondary II Math 10
Unit 4.1
Name ________________________________________________
Period _________
Date _____________________
Expressions 1 Bell Quiz
Combine like terms.
1. 𝟒𝒙𝟐 + 𝟓𝒙𝟐 + 𝒙
2. 𝟕𝒙𝒚 − 𝒙 + 𝟓𝒚 − 𝟗𝒙𝒚 + 𝒚
3. 𝟖 − 𝟑𝒙𝟓 + 𝟔𝒙𝟒 + 𝟏𝟐 + 𝒙𝟓
4. 𝟑𝒙 − 𝟗𝒙 + 𝟑𝒙 + 𝒙 + 𝟏𝟑𝒙
5. 𝟔𝒙𝒚𝟐 − 𝟏𝟓𝒙𝟐 𝒚 + 𝒙𝟐 𝒚 − 𝟒𝒙𝒚𝟐 + 𝒙𝒚𝟐
NUCC | Secondary II Math 11
Unit 4.2
4.2 OPERATIONS WITH POLYNOMIALS
Teacher Notes
Time Frame: Two 40 minute class periods
Materials Needed: Colored pencils, rulers or strips of paper, graphing calculators (optional)
Note: This will be a review lesson for most students, except dividing polynomials.
Launch
Building Polynomial Functions Activity
Reference: http://illuminations.nctm.org/LessonDetail.aspx?id=L282. Encourage students to
work in pairs on the activity sheets. Each student needs the activity sheets, three different
colored pencils, and a strip of paper or a ruler. The discussion generated by questions in the
activity is beneficial.
Students start by identifying a linear function and putting the equation in slope/x-intercept form, y =
m(x - c), where c is the x-intercept. This form serves as a connector with other classes of polynomial
functions and forces students to focus on the x-intercept of the graph. They then choose another
function in the form y = m(x - c) and graph this function on the same axes. Students predict how a
new function, formed by taking the product of the two linear expressions, would appear graphically.
After making their prediction, they graph the resulting quadratic function and compare the actual
function with their prediction. Students can use a graphing utility to check the function formed by
taking the product of the linear factors, but only after making the prediction.
Activity questions that compare the linear functions with the resulting quadratic function focus the
students' attention on the parts of the graphs to be emphasized. Students learn that the quadratic
function has the same x-intercepts as the linear functions, which can be quite a revelation, and that
the y-intercept of the quadratic function is the product of the y-intercepts of the linear functions. In
fact, the y-coordinate of the parabola for a given x-value is always the product of the y-coordinates of
12
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Unit 4.2
the lines for that x-value. Seeing this relationship is easier when x equals 0 and the y-coordinates are
lined up on the y-axis.
Students then use a strip of paper or a ruler to cover parts of the graph. This part of the activity
shows that the sign of the y-coordinate for any point on the parabola can be determined by observing
whether the y-coordinates of the lines for that section of the graph are positive or negative. For
example, if both lines in a section of the graph are above the x-axis, then the parabola will be above
the x-axis, that is, (+) • (+) = (+). If one line in a section of the graph is above the x-axis and the
other is below the x-axis, then the parabola is below the x-axis, that is, (+) • (–) = (–). This result
corresponds to the sign table that students have traditionally used as an aid to graph functions and
inequalities.
Discuss
Have students verbally define a polynomial.

polynomial = a monomial or the sum or difference of monomials. Examples of polynomials: 4, 4
+ d, 𝟓𝒎𝟐 − 𝟐𝒎, k
*
compare and contrast expressions and polynomials
Examples in writing a polynomial
Write a polynomial where each term is a different degree and the polynomial is linear with 2
terms. How about quadratic with 2 terms?
Possible answers: 4x + 9, 𝟒𝒙𝟐 − 𝟗
Using Algebra Tiles
Algebra tiles help connect what is happening with polynomials visually. They also connect geometry
and algebra using area. This website has an excellent PowerPoint on the use of algebra tiles. If no
tiles are available students may just draw in the tile solution.
http://mathbits.com/mathbits/AlgebraTiles/AlgebraTiles.htm
13
Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Unit 4.2
Examples to do with tiles and without tiles:

Adding and Subtracting (putting together like terms!)
𝟐𝒙𝟐 − 𝒙 + 𝒙𝟐 − 𝟓𝒙
𝟑𝒙𝟐 − 𝟒𝒙

𝟑𝒙 − 𝟏 + 𝟐𝒙 + 𝟑𝒙𝟐 + 𝟒
𝟑𝒙𝟐 + 𝟓𝒙 + 𝟑
Multiplying (distributing, (𝒙 + 𝟏)𝟑 𝒑𝒓𝒐𝒃𝒍𝒆𝒎𝒔 𝒍𝒊𝒌𝒆 𝒕𝒉𝒊𝒔 𝒊𝒕 𝒊𝒔 𝒂𝒍𝒔𝒐 𝒄𝒂𝒍𝒍𝒆𝒅 𝒆𝒙𝒑𝒂𝒏𝒅𝒊𝒏𝒈)
𝟒𝒙(𝒙 + 𝟑)
(𝒙 + 𝟑)(𝟐𝒙 + 𝟐)
𝟒𝒙𝟐 + 𝟏𝟐𝒙
𝟐𝒙𝟐 + 𝟐𝒙 + 𝟔𝒙 + 𝟔
𝟐𝒙𝟐 + 𝟖𝒙 + 𝟔
Students will have seen adding, subtracting and multiplying polynomials in Secondary I. We
did not cover dividing polynomials so they may need additional help on this operation. This is
a website that explains dividing polynomials. http://www.youtube.com/watch?v=qd-T-dTtnX4

(𝒙𝟐 + 𝟑𝒙 + 𝟐) ÷ (𝒙 + 𝟏)
Dividing
x+2
𝒙+𝟏
𝒙𝟐 + 𝟑𝒙 + 𝟐
−𝒙𝟐 + 𝒙
2x + 2
Assignment Ready, Set (Operations with Polynomials Worksheet)
14
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© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Unit 4.2
Assignment Go! (Use either option)
Option 1: Polynomial Puzzles Task
For more information and detailed instructions on Polynomial Puzzles see the website. There is also
an overhead sheet to walk the students through (see below). (May be used in Unit 4.3, if not used here)
Reference: http://illuminations.nctm.org/LessonDetail.aspx?id=L798
Polynomial Puzzle Overhead
Polynomial Puzzle Answers for Overhead
Student’s worksheet:
15
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© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Unit 4.2
Option 2: Pascal’s Triangle Task
Polynomial Bell Quiz 1
NUCC | Secondary II Math 16
Unit 4.2
Polynomial Bell Quiz 2
Polynomial Bell Quiz 3
NUCC | Secondary II Math 17
Unit 4.2
Mathematics Content
Cluster Title: Perform arithmetic operations on polynomials.
Standard A.ARP.1: Understand that polynomials form a system analogous to the integers –
namely, they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
Concepts and Skills to Master



Add and subtract polynomials.
Multiply polynomials using the distributive property, and then simplify.
Understand closure of polynomials for addition, subtraction, and multiplication.
Critical Background Knowledge



Understand operations and properties of integers, including closure.
Add and subtract like terms.
Understand the distributive property.
Academic Vocabulary
Like terms, binomial, trinomial, polynomial, closure
Suggested Instructional Strategies


Use algebra tiles or other manipulatives
for addition, subtraction, and
multiplication of polynomials.
Try to find two polynomials whose
sum/product is not a polynomial.
Skills:

Multiply (x2 + 3x – 5)(x + 4) and
determine if the result is a polynomial.
Some Useful Websites:
http://illuminations.nctm.org/LessonDetail.aspx?id=L282
http://mathbits.com/mathbits/AlgebraTiles/AlgebraTiles.htm
http://www.youtube.com/watch?v=qd-T-dTtnX4
http://illuminations.nctm.org/LessonDetail.aspx?id=L798
http://mathforum.org/workshops/usi/pascal/pascal_binomial.html
NUCC | Secondary II Math 18
Unit 4.2
Building Polynomial Functions
Name ________________________________________________
Period _________
1. What is the equation of the linear function shown to the right?
2. How did you find it?
3. The slope – y-intercept form of a linear function is y = mx + b.
If you’ve written the equation in another form, rewrite your equation in slope – y-intercept
form.
4. Now, factor out the slope, and rewrite the function y = m ( x + b ) .
as
m
5. Choose a second linear function and write it in slope – y-intercept form.
6. Graph the function on the axis above, and be sure to label it.
7. Rewrite your second function with the slope factored out (just like you did in Question 4).
8. For each function, what does
represent on the graph?
m
b
y = m ( x − c ) could be called the slope – x-intercept form of a linear
If you let c =m− b , then the
form
equation, where c is the x-intercept. The factor theorem states that if c is a root (x-intercept) of a
polynomial function, then ( x − c) must be a factor of that polynomial function. Note that ( x − c) is a
factor of the expression. The only other factor is the slope m.
9. From their slope – y-intercept form, multiply the two functions together.
19
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© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Unit 4.2
10. Graph the resulting function on the same axis as the two lines on the previous page.
11. What kind of function did you get?
12. What relationship do you see between the graph from Question 10 and the lines?
• …and the x-intercepts?
• …and the y-intercepts?
13. Identify the left-most x-intercept on the graph. With a straight-edge, cover everything to the
right of that point. What connections do you see relating the signs of the y-values?
14. Identify the right-most intercept on the graph. With a straight-edge, cover up everything to the left
of that point. What connections do you see relating to the signs of the y-values?
Complete the following sentences.
15. When both lines are above the x-axis, the y-values
are
parabola
.
and the
16. When both lines are below the x-axis, the y-values
are
parabola
.
and the
17. When one line is above the x-axis and the other is below the x-axis, the parabola
.
y-VALUE OF L1
y-VALUE OF L2
+
+
-
+
+
-
PARABOLA IS
ABOVE/BELOW
THE x-AXIS
20
Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Unit 4.2
18. Based on the patterns you saw on the previous page, draw a sketch of the quadratic function that
would be obtained from the linear expressions of these lines.
19. Write the equation for each line.
20. To check your sketch in Question 18, multiply the expressions together, and graph the resulting
function on the grid above. How accurate was your sketch?
21
Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics
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Unit 4.2
Operations with Polynomials Worksheet
Name ________________________________________________
Period _________
Anatomy of a Polynomial
Identify the following terms from the polynomial, 𝒙𝟐 − 𝒙 + 𝟔 .
1. variable(s)
2. coefficient(s)
3. operation(s)
Write a polynomial where each term is a different degree and the polynomial satisfies each condition.
4. linear polynomial with 1 term
5. linear polynomial with 2 terms
6. quadratic polynomials with 3 terms
7. cubic polynomials with 4 terms
Use Algebra Tiles to perform the following operations. Draw the algebra tile solution and then write
the solution.
8. 5𝑥 2 − 4𝑥 + 𝑥 2 − 2𝑥
9. 3𝑥 − 6 + 6𝑥 − 𝑥 2 + 4
10. 3𝑥(2𝑥 + 3)
11. 2𝑥(4𝑥 + 1)
12. (𝑥 + 1)(𝑥 + 3)
13. (𝑥 + 2)(2𝑥 + 1)
14. (2𝑥 2 + 4𝑥 + 3) ÷ (𝑥 + 1)
15. (4𝑥 2 + 6𝑥 + 2) ÷ (2𝑥 + 2)
NUCC | Secondary II Math 22
Unit 4.2
Adding and Subtracting Polynomials
16. 2𝑎2 + 5𝑎2 + 𝑎
17. 4𝑚𝑛 − 3𝑚 + 2𝑛 − 9𝑚𝑛 + 𝑛
18. 6 − 𝑘 5 + 4𝑘 4 + 9 + 𝑘 5
19. 2𝑥 − 9𝑥 + 4𝑥 + 𝑥 + 2𝑥
20. 𝑎𝑏 − 7𝑎𝑏 + 𝑎𝑏 + 8𝑎𝑏 2
21. 4𝑥𝑦 2 − 9𝑥 2 𝑦 + 𝑥 2 𝑦 + 7𝑥𝑦 2 + 𝑥𝑦 2
Multiplying Polynomials
22. 8𝑚𝑛(5𝑚 + 3𝑛 − 6)
23. 3𝑥𝑦 2 (4𝑥 − 7𝑥𝑦)
24. (3𝑥 − 1)(𝑥 + 6)
25. (𝑥 − 7)(𝑥 + 7)
26. 𝑥(𝑥 + 2)(𝑥 − 5)
27. 2𝑥(𝑥 − 10)(𝑥 2 − 4)
28. (𝑥 + 3)3
29. (2𝑥 − 1)4
Dividing Polynomials
30. (6𝑚𝑛2 − 3𝑚𝑛 + 2𝑥 2 𝑦) ÷ (𝑥𝑦)
31. (𝑘 2 − 10𝑘 − 24) ÷ (𝑘 + 2)
32. (𝑥 3 + 𝑦 3 ) ÷ (𝑥 + 𝑦)
33. (𝑡 2 + 8𝑡 + 15)(𝑡 + 3)−1
34. (𝑥 3 + 3𝑥 2 − 7𝑥 − 21)(𝑥 + 3)−1
35. (60𝑚2 − 98𝑛2 ) ÷ (10𝑚 + 14𝑛)
Prove each Identity
36. 𝑥 2 − 𝑦 2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
37. (𝑥 + 𝑦)2 = 𝑥 2 + 2𝑥𝑦 + 𝑦 2
NUCC | Secondary II Math 23
Unit 4.2
Polynomial Puzzler
Name ________________________________________________
Period _________
Fill in the empty spaces to complete the puzzle. In any row, the two left spaces should multiply
to equal the right-hand space. In any column, the two top spaces should multiply to equal the
bottom space.
1.
2.
3.
4.
5.
6.
24
Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
Unit 4.2
Pascal’s Triangle Task
Group Members ________________________________________________
Period _____
A. Determine the pattern and then complete rows 5, 6 and 7 of the triangle.
Row
Number
Pascal’s Triangle
0
1
1
1
2
1
3
1
4
___
___
6
7
___
4
___
___
___
2
3
1
5
1
3
6
___
___
___
1
4
___
___
___
1
1
___
___
___
___
___
___
___
___
___
B. Multiply the following binomials.
1. (𝑥 + 𝑦)0
2. (𝑥 + 𝑦)1
3. (𝑥 + 𝑦)2
4. (𝑥 + 𝑦)3
5. (𝑥 + 𝑦)4
NUCC | Secondary II Math 25
Unit 4.2
C. What is the connection between the expanded binomials and the Pascal’s Triangle?
D. Expand (multiply) the binomials using Pascal’s Triangle.
1. (𝑥 + 𝑦)6
2. (𝑥 + 𝑦)7
3. (𝑥 + 𝑦)8
E. Prepare a 2 minute presentation about Pascal’s triangles and binomial expansions. Write your presentation
here and be creative.
Reference: http://mathforum.org/workshops/usi/pascal/pascal_binomial.html
Extension: How can we use Pascal's triangle to write the expansion of any binomial (x+y)n.
NUCC | Secondary II Math 26
Unit 4.2
Name ________________________________________________
Period _________
Date _____________________
Polynomial 1 Bell Quiz
Write an expression for each statement.
1.
Six lemons and 4 pickles
2.
The quotient of the sum of five and a
number and nine
Write a statement for the expression.
3.
𝟐 + 𝒙𝟓
4.
5.
𝒎−𝟒
𝟑
Write an expression with 3 different operations where you begin with the
number 5 and end with 23.
NUCC | Secondary II Math 27
Unit 4.2
Name ________________________________________________
Period _________
Date _____________________
Polynomial 2 Bell Quiz
Identify the following terms from the polynomial 𝟑𝒙𝟐 − 𝟒𝒙 + 𝟏𝟑.
1. Constant(s)
2. Coefficient(s)
3. Degree
Write a polynomial where each term is a different
degree and the polynomial satisfies each condition.
4. Linear polynomial with 1 term
5. Quadratic polynomial with 3 terms
NUCC | Secondary II Math 28
Unit 4.2
Name ________________________________________________
Period _________
Date _____________________
Polynomial 3 Bell Quiz
Multiply the polynomials.
1.
𝟐𝒙(𝒙𝟐 + 𝟓)
2.
𝟑𝒙𝒚(𝒙 + 𝟓𝒚 − 𝟐𝒙𝒚)
3.
(𝒙 + 𝟓)(𝒙 + 𝟐)
4.
(𝟐𝒙 − 𝟑)(𝒙 + 𝟏)
5.
𝟒𝒙(𝒙 + 𝟑)(𝟑𝒙 − 𝟐)
NUCC | Secondary II Math 29
Unit 4.3
4.3 FACTORING POLYNOMIALS
Teacher Notes
Time Frame: Two 40-minute class periods
Materials Needed:
Launch:
Number Factor Relay
Divide the room in half and give each student a number. Give the other side of the
room the same numbers. You can set up more teams if you want, however, 2 is
usually good. For the relay you will put a number to be factored on the whiteboard
and then call out a number so that both students with that number come to the board
to factor. It is a race to see who can write all the factors the quickest. There may be
more than one possible answer. The winning team gets a point. If neither student can
figure out the factors call another number up to assist them. 10 minutes should get
through a class.
Example: Write the number 38 on board, call out number 4, number 4 students come
to the board and write 2 and 19.
*Students should know factors.
Discuss
Ready
A. Greatest Common Factor (Students should know this, but may need reminded. Show them the
method below, but most students can pull out the greatest common factors by observation.)
Ex: 4𝑥 2 + 20𝑥
Write the factors for each term and circle the factors in common. The greatest common factor
goes in front of the parentheses and the remaining numbers stay inside the parentheses.
2∙2∙𝑥∙𝑥
𝟒𝒙(𝒙 + 𝟓)
2∙2∙5∙𝑥
More Examples: 14𝑚 − 21𝑚3 𝑛
𝟕𝒎(𝟐 − 𝟑𝒎𝟐 𝒏)
18𝑡𝑟 2 + 9𝑟
𝟗𝒓(𝟐𝒕𝒓 + 𝟏)
3𝑥 2 − 6𝑥 + 12𝑥 3
𝟑𝒙(𝒙 − 𝟐 + 𝟒𝒙𝟐 )
*Remind students to always look to see if the terms of the polynomial have a common factor
first!!!!
NUCC | Secondary II Math 30
Unit 4.3
Set
B. Trinomials (Students may or may not have learned this. Experience is usually minimal.)
1. Leading coefficient of 1
Vocab Reminders
trinomial = polynomial with 3 terms. A type of quadratic equation.
leading coefficient = the number multiplied to the variable of the first term of a polynomial
in standard form.
Ex: 2𝑥 2 + 𝑥 + 3 is a trinomial and the leading coefficient is 2
Procedure: Use 2 factors of the constant term that you can add to get the coefficient of
the second term.
Ex: 𝑥 2 + 6𝑥 + 8 factors are (𝑥 + 4)(𝑥 + 2)
Note 4 ∙ 2 = 8 𝑎𝑛𝑑 4 + 2 = 6
*Do one example of multiplying the factors back out to get the trinomial as a check.
More examples for student practice:
𝑥 2 − 2𝑥 − 8
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (𝑥 − 4)(𝑥 + 2) , − 4 ∙ 2 = −8 𝑎𝑛𝑑 − 4 + 2 = −2
𝑥 2 − 6𝑥 + 8
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (𝑥 − 4)(𝑥 − 2) , − 4 ∙ −2 = 8 𝑎𝑛𝑑 − 4 + (−2) = −6
𝑥2 − 𝑥 − 6
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (𝑥 − 3)(𝑥 + 2) , − 3 ∙ 2 = −6 𝑎𝑛𝑑 − 3 + 2 = −1
𝑥 2 + 5𝑥 − 6
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (𝑥 − 6)(𝑥 + 1) , − 6 ∙ 1 = −6 𝑎𝑛𝑑 − 6 + 1 = 5
𝑥 2 + 𝑥 − 56
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (𝑥 − 7)(𝑥 + 8) , − 7 ∙ 8 = −56 𝑎𝑛𝑑 − 7 + 8 = 1
Extra explanation at: http://www.regentsprep.org/regents/math/algebra/AV6/Ltri1.htm
2. Leading coefficients other than 1:
Procedure:
a. Multiply the leading coefficient by the constant term.
b. Multiply to get the constant term and add those same 2 numbers to get the second
term coefficient.
c. Divide the leading coefficient back out from the numerical part of the factors and
simplify.
Ex: 3𝑥 2 + 10𝑥 + 8
a. 3 ∙ 8 = 24 𝑠𝑜 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑡𝑟𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑖𝑠 𝑥 2 + 10𝑥 + 24
b. factors of adjusted trinomial are (𝑥 + 4)(𝑥 + 6)
4
6
4
c. (𝑥 + 3)(𝑥 + 3) (You can leave the (𝑥 + 3) as it is easier to find the zeros, but
tradition moves the denominator in front of the x)
NUCC | Secondary II Math 31
Unit 4.3
(3𝑥 + 4)(𝑥 + 2) (multiply this one back out to check work)
More examples for student practice:
5𝑥 2 + 13𝑥 − 6 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (5𝑥 − 2)(𝑥 + 3)
4𝑥 2 + 19𝑥 + 15 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (𝑥 + 1)(4𝑥 + 15)
3 𝑥 2 − 2𝑥 − 8 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (3𝑥 + 4)(𝑥 − 2)
9 𝑥 2 − 15𝑥 + 4 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 (3𝑥 − 4)(3𝑥 − 1)
Go! (Do one or both options for practice)
Option 1: Polynomial Factoring Practice Worksheet
NUCC | Secondary II Math 32
Unit 4.3
Option 2: Polynomial Puzzle
For more information and detailed instructions on Polynomial Puzzles see the website. There is
also an overhead sheet to walk the students through (see below). See Unit 4.2 for pages to copy.
http://illuminations.nctm.org/LessonDetail.aspx?id=L798
Polynomial Puzzle Overhead
Polynomial Puzzle Answers for Overhead
Student’s worksheet:
NUCC | Secondary II Math 33
Unit 4.3
Factoring Bell Quiz 1
NUCC | Secondary II Math 34
Unit 4.3
Mathematics Content
Cluster Title: Write expressions in equivalent forms to solve problems.
Standard A.SSE.3: Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.★
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value
of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. (For
example the expression 1.15t can be rewritten as (1.151/12)12t - 1.01212t to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.)
Concepts and Skills to Master
•
•
Rewrite expressions in different forms using mathematical properties.
Given a context determine the best form of an expression.
Critical Background Knowledge
•
•
Understand the distributive property in simplifying and expanding expressions.
Various types of factoring skills.
Academic Vocabulary
factors, coefficients, terms, exponent, base, constant, variable, binomial, monomial, polynomial
Suggested Instructional Strategies
•
•
Connect point-slope form to
transformation of a line.
Connect to the forms of a quadratic
function.
Skills:
•
•
•
•
Given a quadratic in standard form,
rewrite in vertex form and list the
properties used in each step.
One of the factors of 0.2x3 – 1.2x2 -0.6x
is (x-2). Find the other factors.
Find multiple ways to rewrite x6 – y6.
2
Rewrite in radical form 𝑥 ⁄3 in radical
form.
Some Useful Websites:
http://illuminations.nctm.org/LessonDetail.aspx?id=L798
http://www.regentsprep.org/regents/math/algebra/AV6/Ltri1.htm
NUCC | Secondary II Math 35
Unit 4.3
Name ________________________________________________
Period _________
Polynomial Factoring Practice
Factor each polynomial completely.
1. 𝟔𝟒𝐚𝟐 𝐛𝟑 − 𝟏𝟔𝐛𝟐 𝐚𝟑
2. 𝐱 𝟐 + 𝟐𝟎𝐱 + 𝟏𝟎𝟎
3. 𝟑𝐱 𝟐 − 𝟐𝐱 − 𝟓
4. 𝟐𝐧𝟐 + 𝟓𝐧 + 𝟐
5. 𝟏𝟒𝟒𝐱 𝟐 − 𝟏𝟎𝟖𝐲 𝟐 − 𝟔𝟎𝐳 𝟐
6. 𝟑𝐰 𝟐 − 𝟖𝐰 + 𝟒
7. 𝟓𝐦𝟑 𝐧 + 𝟏𝟓𝐦𝐧𝟐
8. 𝟐𝐦𝟐 + 𝟏𝟏𝐦 + 𝟓
9. 𝟓𝐧𝟐 − 𝟏𝟖𝐧 + 𝟗
10. 𝟔𝐱 𝟑 𝐲 + 𝟏𝟐𝐱𝐲
11. 𝟏𝟓𝐳 𝟐 − 𝟐𝟕𝐳 − 𝟔
12. 𝐦𝟐 − 𝟕𝐦 + 𝟏𝟐
13. 𝟒𝐭 𝟐 − 𝟏𝟓𝐭 − 𝟐𝟓
14. 𝐤 𝟐 − 𝟓𝐤 − 𝟑𝟔
15. 𝟏𝟓𝐱 𝟐 − 𝟐𝟕𝐱 − 𝟔
16. 𝟑𝟒𝐱 𝟐 𝐲𝐳 𝟑 − 𝟏𝟕𝐱𝐲𝐳
17. 𝐤 𝟐 + 𝟏𝟏𝐤 + 𝟏𝟖
18. 𝟐𝟖𝐦𝐧𝟑 − 𝟏𝟒𝐦𝟐 𝐧
19. 𝐳 𝟐 − 𝟏𝟑𝐳 + 𝟑𝟔
20. 𝟒𝐱 𝟐 − 𝟑𝟓𝐱 + 𝟒𝟗
21. 𝟔𝐦𝟐 + 𝟑𝟕𝐦 + 𝟔
22. 𝟏𝟖𝐱𝐲 𝟑 − 𝟔𝐱𝐲 𝟐
23. 𝟔𝟗 + 𝟐𝟔𝐯 𝟑 − 𝟓𝟐𝐯
24. 𝐱 𝟐 − 𝐱 − 𝟑𝟎
Write a polynomial that has the given factors.
25. (𝟐𝐱 − 𝟑)(𝟒𝐱 + 𝟔)
26. 𝟕𝐱𝐲(𝟐𝐱 − 𝟗𝐱𝐲 𝟐 )
27. (𝐱 − 𝟔)(𝐱 + 𝟕)
NUCC | Secondary II Math 36
Unit 4.4
4.4 FACTORING SPECIAL PRODUCTS
Teacher Notes
Time Frame: Two 40-minute class periods
Materials Needed:
Launch: (15 minutes)
Patterns with Polynomial Products
Have the students work in groups to multiply out the sets of polynomials, there are 5
different sets, so some groups will be doing the same discovery. Tell the students to
look for patterns in the sets (factors) and in the product. Students will be asked to
present their discoveries to the class. The patterns students should discover are in the
boxes to the right.
NUCC | Secondary II Math 37
Unit 4.4
Discuss:
1. Difference of Two Squares
a. (𝑥 + 4)(𝑥 − 4)
b. (𝑥 + 2)(𝑥 − 2)
Patterns
Factor: One factor is adding and one is minus.
The first letter is the same and the last
number are the same for each factor.
c. (𝑥 + 5)(𝑥 − 5)
d. (𝑥 + 7)(𝑥 − 7)
e. (𝑥 + 10)(𝑥 − 10)
Product: Always subtracting
The first letter and the last are squared.
The middle terms cancel out
2. Perfect Square Sums
a. (𝑥 + 4)(𝑥 + 4)
b. (𝑥 + 2)(𝑥 + 2)
Patterns
Factor: Both factors are adding.
Both factors are identical and could be
written (𝑥 + 4)2
c. (𝑥 + 5)(𝑥 + 5)
d. (𝑥 + 7)(𝑥 + 7)
e. (𝑥 + 10)(𝑥 + 10)
3. Perfect Square Differences
a. (𝑥 − 4)(𝑥 − 4)
b. (𝑥 − 2)(𝑥 − 2)
c. (𝑥 − 5)(𝑥 − 5)
d. (𝑥 − 7)(𝑥 − 7)
e. (𝑥 − 10)(𝑥 − 10)
Product: The first and last terms are the squares
of the first and last terms of the binomial
The middle term is twice the product of
the two terms in the binomial
Only the adding operation in the product
Patterns
Factor: Both factors are subtracted
Both factors are identical and could be
written (𝑥 − 4)2
Product: The first and last terms are the
of the first and last terms of the
binomial
The middle term is twice the product of
the two terms in the binomial
Operations of the product are always
subtraction then addition
NUCC | Secondary II Math 38
Unit 4.4
4. Sum of Two Cubes
a. (𝑥 + 4)(𝑥 2 − 4𝑥 + 16)
b. (𝑥 + 2)(𝑥 2 − 2𝑥 + 4)
c. (𝑥 + 1)(𝑥 2 − 𝑥 + 1)
d. (𝑥 + 3)(𝑥 2 − 3𝑥 + 9)
Patterns
Factor: Factors are binomial & trinomial
Binomial is a sum
Trinomial is subtracting then adding
Middle term coefficient of trinomial is
same as constant of the binomial
Constant of trinomial is the square of the
constant of the binomial
Product: A sum
The terms are each cubed
5. Difference of Two Cubes
a. (𝑥 − 4)(𝑥 2 + 4𝑥 + 16)
b. (𝑥 − 2)(𝑥 2 + 2𝑥 + 4)
c. (𝑥 − 1)(𝑥 2 + 𝑥 + 1)
d. (𝑥 − 3)(𝑥 2 + 3𝑥 + 9)
Patterns
Factor: Factors are binomial & trinomial
Binomial is subtraction
Trinomial is addition
Middle term coefficient of trinomial is
same as constant of the binomial
Constant of trinomial is the square of the
constant of the binomial
Product: A difference
The terms are each cubed
Ready
Conjugates = Two binomials with the same two terms but opposite signs separating the terms are
called conjugates of each other. Following are examples of conjugates:
Which special product has conjugates? Difference of 2 Squares
NUCC | Secondary II Math 39
Unit 4.4
Have students make the following table in their notes and write examples from the leading task in
the example column.
SPECIAL PRODUCT
FACTORING PATTERNS
EXAMPLE
POLYNOMIALS
Difference of Two Squares
𝒂𝟐 − 𝒃𝟐
(𝒂 + 𝒃)(𝒂 − 𝒃)
Perfect Square Sums
𝒂𝟐 + 𝟐𝒂𝒃 + 𝒃𝟐
(𝒂 + 𝒃)𝟐
Perfect Square Differences
𝒂𝟐 − 𝟐𝒂𝒃 + 𝒃𝟐
(𝒂 − 𝒃)𝟐
Sum of Two Cubes
(𝒂 + 𝒃)(𝒂𝟐 − 𝒂𝒃 + 𝒃𝟐 )
𝒂𝟑 + 𝒃𝟑
Difference of Two Cubes
(𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐 )
𝒂𝟑 − 𝒃𝟑
Set
Practice factoring a few of each of the special products.
1. 𝑥 2 − 25
(𝑥 + 5)(𝑥 − 5)
2. 𝑥 2 − 81
(𝑥 + 9)(𝑥 − 9)
3. 𝑥 2 − 49
(𝑥 + 7)(𝑥 − 7)
4. 𝑥 2 + 12𝑥 + 36
5. 𝑥 2 + 4𝑥 + 4
(𝑥 + 6)2
(𝑥 + 2)2
NUCC | Secondary II Math 40
Unit 4.4
6. 𝑥 2 − 20𝑥 + 100
7. 𝑥 2 − 10𝑥 + 25
(𝑥 − 10)2
(𝑥 − 5)2
8. 𝑥 3 + 8
(𝑥 + 2)(𝑥 2 − 2𝑥 + 4)
9. 𝑥 3 + 1
(𝑥 + 1)(𝑥 2 − 𝑥 + 1)
10. 𝑥 3 − 27
(𝑥 − 3)(𝑥 2 + 3𝑥 + 9)
11. 𝑥 3 − 64
(𝑥 − 4)(𝑥 2 + 4𝑥 + 16)
Go! (Do as many options as needed for your students)
Option 1. Special Products Factoring Worksheet
NUCC | Secondary II Math 41
Unit 4.4
Option 2. Special Products and Factoring Task
(*Similar activity as the opening task. There are six pages to this task.)
NUCC | Secondary II Math 42
Unit 4.4
Option 3. I Have, Who Has Factoring Worksheet
The game is called “I have, who has factoring” It is a review game for factoring trinomials.
(reviews factoring from units 4.1 through 4.4) This is a game that is found at
http://www.ilovemath.org/index.php?option=com_docman&task=doc_details&gid=100
NUCC | Secondary II Math 43
Unit 4.4
Special Products Bell Quiz 1
NUCC | Secondary II Math 44
Unit 4.4
Mathematics Content
Cluster Title: Interpret the structure of expressions.
Standard A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For
example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be
factored as (x2 – y2)(x2 + y2).
Concepts and Skills to Master
•
•
•
Understand that an expression has different forms.
Justify the different forms based on mathematical properties.
Interpret different symbolic notation.
Critical Background Knowledge
•
•
Understand the distributive property in simplifying and expanding expressions.
Various types of factoring skills.
Academic Vocabulary
factors, coefficients, terms, exponent, base, constant, variable, binomial, monomial, polynomial
Suggested Instructional Strategies
• This standard should be taught in
conjunction with standard A.DDE.3, with
heavy emphasis on justification.
Skills:
• Factor x6 – y6 as the difference of two
squares and as the difference of cubes.
Justify that the resulting expressions are
equivalent.
Some Useful Websites:
Distributing and factoring using area:
http://illuminations.nctm.org/LessonDetail.aspx?id=L744
Difference of squares:
http://illuminations.nctm.org/LessonDetail.aspx?id=L276
NUCC | Secondary II Math 45
Unit 4.4
Patterns with Polynomial Products
Have the students work in groups to multiply out the sets of polynomials, there are 5
different sets of polynomials, so some groups will be doing the same discovery. Tell the
students to look for patters in the sets (factors) and in the product. Students will be asked to
present their discoveries to the class. Cut the sets apart to give to the groups.
a. (𝐱 + 𝟒)(𝐱 − 𝟒)
a. (𝒙 + 𝟒)(𝒙 + 𝟒)
b. (𝐱 + 𝟐)(𝐱 − 𝟐)
b. (𝒙 + 𝟐)(𝒙 + 𝟐)
c. (𝐱 + 𝟓)(𝐱 − 𝟓)
c. (𝒙 + 𝟓)(𝒙 + 𝟓)
d. (𝐱 + 𝟕)(𝐱 − 𝟕)
d. (𝒙 + 𝟕)(𝒙 + 𝟕)
e. (𝐱 + 𝟏𝟎)(𝐱 − 𝟏𝟎)
e. (𝒙 + 𝟏𝟎)(𝒙 + 𝟏𝟎)
a. (𝒙 − 𝟒)(𝒙 − 𝟒)
a. (𝒙 + 𝟒)(𝒙𝟐 − 𝟒𝒙 + 𝟏𝟔)
b. (𝒙 − 𝟐)(𝒙 − 𝟐)
b. (𝒙 + 𝟐)(𝒙𝟐 − 𝟐𝒙 + 𝟒)
c. (𝒙 − 𝟓)(𝒙 − 𝟓)
c. (𝒙 + 𝟏)(𝒙𝟐 − 𝒙 + 𝟏)
d. (𝒙 − 𝟕)(𝒙 − 𝟕)
d. (𝒙 + 𝟑)(𝒙𝟐 − 𝟑𝒙 + 𝟗)
e. (𝒙 − 𝟏𝟎)(𝒙 − 𝟏𝟎)
a. (𝒙 − 𝟒)(𝒙𝟐 + 𝟒𝒙 + 𝟏𝟔)
b. (𝒙 − 𝟐)(𝒙𝟐 + 𝟐𝒙 + 𝟒)
c. (𝒙 − 𝟏)(𝒙𝟐 + 𝒙 + 𝟏)
d. (𝒙 − 𝟑)(𝒙𝟐 + 𝟑𝒙 + 𝟗)
NUCC | Secondary II Math 46
Unit 4.4
Special Products Factoring Worksheet
Name _______________________________________________
Period ___________
Factor each polynomial completely.
1. 𝒙𝟐 − 𝟏𝟗𝟔
2. 𝒙𝟐 − 𝟏𝟔
3. 𝒙𝟑 + 𝟏𝟐𝟓
4. 𝐱 𝟐 + 𝟏𝟒𝐱 + 𝟒𝟗
5. 𝐱 𝟐 + 𝟑𝟎𝐱 + 𝟐𝟐𝟓
6. 𝐱 𝟐 − 𝟏𝟖𝐱 + 𝟖𝟏
7. 𝐱 𝟐 − 𝟐𝟒𝐱 + 𝟏𝟒𝟒
8. 𝐱 𝟐 + 𝟔𝐱 + 𝟗
9. 𝐱 𝟑 + 𝟏
10. 𝐱 𝟑 − 𝟐𝟏𝟔
11. 𝐱 𝟑 − 𝟐𝟕
12. 𝐱 𝟐 + 𝟏𝟒𝐱 + 𝟒𝟗
13. 𝐱 𝟐 − 𝟐𝟓
14. 𝒙𝟐 − 𝟖𝟏
15. 𝒙𝟑 − 𝟓𝟏𝟐
16. 𝒙𝟐 + 𝒙 + 𝟏
17. 𝒙𝟐 + 𝟒𝒙 + 𝟒
18. 𝒙𝟐 − 𝟐𝟐𝒙 + 𝟏𝟐𝟏
19. 𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
20. 𝒙𝟑 + 𝟏𝟎𝟎𝟎
21. 𝟖𝒙𝟑 + 𝟏
22. 𝟐𝟓𝒙𝟐 − 𝟔𝟒
23. 𝒙𝟐 − 𝟏𝟔𝒚𝟐
24. 𝒙𝟑 − 𝒚𝟑
25. 𝒙𝟐 + 𝟖𝒙 + 𝟏𝟔
26. 𝒙𝟐 + 𝟒𝟎𝒙 + 𝟒𝟎𝟎
27. 𝒙𝟐 − 𝟐𝟔𝒙 + 𝟏𝟔𝟗
28. 𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
29 𝟔𝟒𝒙𝟑 + 𝟖
30. 𝒙𝟐 − 𝒚𝟐
31. 𝟖𝒙𝟑 − 𝟐𝟕
32. 𝟗𝒙𝟐 − 𝟐𝟓
33. 𝒙𝟒 − 𝟏𝟔
NUCC | Secondary II Math 47
Unit 4.4
34. Given that the area of a square is 𝟑𝟔𝒙𝟐 − 𝟒𝟗, write and expression for square’s length
and width?
35. Given that the area of a square is 𝒙𝟐 + 𝟏𝟔𝒙 + 𝟔𝟒, write and expression for square’s
length and width?
36. Given that the area of a square is 𝒙𝟐 − 𝟑𝟐𝒙 + 𝟐𝟓𝟔, write and expression for square’s
length and width?
37. A company makes square copper tiles with an area of 𝒙𝟐 + 𝟐𝟒𝒙 + 𝟏𝟒𝟒. Write an
expression for the perimeter of a tile.
38. A company makes square iron sheets with an area of 𝒙𝟐 + 𝟔𝒙 + 𝟗.
Write an expression for the perimeter of the iron sheets.
39. Write your own problem and its solution using special products.
NUCC | Secondary II Math 48
Unit 4.4
Special Products and Factoring Task
Names in Group _______________________________________________________________
Date ________________________________
1.
Period ___________
As a group multiply out the following binomials.
a. Each student will multiply one set of factors and find the product to complete the
table.
Student
Factors
Product
(x + 2)(x – 2)
(x – 5 )(x + 5)
(x + 3)(x – 3)
(2x + 1)(2x – 1)
(3x – 2 )(3x + 2)
b. Write 3 observation or patterns that you see regarding the factors and/or their
product. Be prepared to share your observations with the class.
NUCC | Secondary II Math 49
Unit 4.4
c. Using your observations, factor the product to complete the table.
Student
Product
Factors
𝒙𝟐 − 𝟒𝟗
𝒙𝟐 − 𝟏𝟔
𝒙𝟐 − 𝟏
𝟐𝟓𝒙𝟐 − 𝟔𝟒
𝟒𝒙𝟐 − 𝟏𝟎𝟎
d. Why might this special product be called the difference of 2 squares?
e.
Using your observations, write a formula for the difference of 2 squares.
2. As a group multiply out the squared binomial.
a. Each student will multiply one squared binomial and find the product to complete
the table.
Student
Factors (Squared
Binomial)
Product
(𝒙 + 𝟐)𝟐
(𝒙 + 𝟓)𝟐
(𝒙 + 𝟏)𝟐
(𝟐𝒙 + 𝟑)𝟐
(𝟑𝒙 + 𝟏)𝟐
NUCC | Secondary II Math 50
Unit 4.4
b. Write 3 observation or patterns that you see regarding the factors and/or their
product. Be prepared to share your observations with the class.
c. Using your observations, factor the product to complete the table.
Student
Product
Factors (squared binomial)
𝒙𝟐 + 𝟔𝒙 + 𝟗
𝒙𝟐 + 𝟖𝒙 + 𝟏𝟔
𝒙𝟐 + 𝟏𝟒𝒙 + 𝟒𝟗
𝟐𝟓𝒙𝟐 + 𝟖𝟎𝒙 + 𝟔𝟒
𝟒𝒙𝟐 + 𝟒𝟎𝒙 + 𝟏𝟎𝟎
d. Why might this special product be called the square of a binomial?
e.
Using your observations, write a formula for the square of a binomial with a sum.
NUCC | Secondary II Math 51
Unit 4.4
3. As a group multiply out the squared binomial.
a. Each student will multiply one squared binomial and find the product to complete
the table.
Student
Factors (Squared
Binomial)
Product
(𝒙 − 𝟐)𝟐
(𝒙 − 𝟔)𝟐
(𝒙 − 𝟒)𝟐
(𝟐𝒙 − 𝟑)𝟐
(𝟑𝒙 − 𝟐𝟓)𝟐
b. Write 3 observation or patterns that you see regarding the factors and/or their
product. Be prepared to share your observations with the class.
c. Using your observations, factor the product to complete the table.
Student
Product
Factors (squared binomial)
𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
𝒙𝟐 − 𝟐𝟎𝒙 + 𝟏𝟎𝟎
𝒙𝟐 − 𝟏𝟒𝒙 + 𝟒𝟗
𝟗𝒙𝟐 − 𝟐𝟒𝒙 + 𝟏𝟔
𝟒𝒙𝟐 − 𝟑𝟔𝒙 + 𝟖𝟏
NUCC | Secondary II Math 52
Unit 4.4
d. How is this special product different from the last square of a binomial?
e.
Using your observations, write a formula for the square of a binomial with a
difference.
5. As a group multiply out the factors. Each student will multiply one factor and find the
product to complete the table.
Student
Factors (Cubed
Binomial)
Product
(𝒙 + 𝟐)(𝒙𝟐 − 𝟐𝒙 + 𝟒)
(𝒙 + 𝟓)(𝒙𝟐 − 𝟓𝒙 + 𝟐𝟓)
(𝒙 + 𝟑)(𝒙𝟐 − 𝟑𝒙 + 𝟗)
(𝟐𝒙 + 𝟑)(𝟒𝒙𝟐 − 𝟔𝒙 + 𝟗)
(𝟑𝒙 + 𝟏)(𝟗𝒙𝟐 − 𝟑𝒙 + 𝟏)
6. Write 3 observations or patterns that you see regarding the factors and/or their
products. Be prepared to share your observations with the class.
NUCC | Secondary II Math 53
Unit 4.4
7. Using your observations, factor the product to complete the table.
Student
Product
Factors
𝒙𝟑 + 𝟖
𝒙𝟑 + 𝟔𝟒
𝒙𝟑 + 𝟏𝟎𝟎𝟎
𝟐𝟕𝒙𝟑 + 𝟏𝟐𝟓
𝟔𝟒𝒙𝟑 + 𝟏
8.
Using your observations, write a formula for the sum of two cubes.
9. What is a conjugate? Give the definition and 2 examples. Did any of the patterns you
discovered above have conjugates? Which ones?
NUCC | Secondary II Math 54
Unit 4.4
I Have, Who Has Factoring Worksheet
I have
x2 – 18x + 81.
Who has
(x – 10)(x + 1)?
I have
x2 – 9x – 10.
Who has
(x + 12)(x – 3)?
I have
x2 – 9x – 36.
Who has
(x + 7)(x + 7)?
I have
x2 + 14x + 49.
Who has
(x + 2)(x – 6)?
I have
x2 – 4x – 12.
Who has
(x + 11)(x – 3)?
I have
x2 + 8x – 33.
Who has
(x + 8)(x + 4)?
I have
x2 + 12x + 32.
Who has
(x – 15)(x – 4)?
I have
x2 – 19x + 60.
Who has
(x – 6)(x + 6)?
I have
x2 – 36.
Who has
(x + 2)(x + 7)?
I have
x2 + 9x + 14.
Who has
(x + 9)(x – 5)?
I have
x2 + 4x – 45.
Who has
(x + 1)(x + 1)?
I have
x2 + 2x + 1.
Who has
(x + 9)(x + 9)?
I have
x2 + 18x + 81.
Who has
(x + 6)(x – 10)?
I have
x2 – 4x – 60.
Who has
(x – 6)(x – 6)?
I have
x2 – 12x + 36.
Who has
(x – 15)(x + 4)?
I have
x2 – 11x – 60.
Who has
(x + 1)(x + 10)?
I have
x2 + 11 x + 10.
Who has
(x – 9)(x – 3)?
I have
x2 – 12x + 27.
Who has
(x – 6)(x + 4)?
NUCC | Secondary II Math 55
Unit 4.4
I have
x2 – 2x – 24.
Who has
(x – 8)(x – 4)?
I have
x2 – 12x + 32.
Who has
(x – 9)(x + 6)?
I have
x2 – 3x – 54.
Who has
(x + 15)(x + 4)?
I have
x2 + 19x + 60.
Who has
(x + 7)(x + 3)?
I have
x2 + 10x + 21.
Who has
(x – 8)(x – 3)?
I have
x2 – 11x + 24.
Who has
(x + 5)(x + 9)?
I have
x2 + 14x + 45.
Who has
(x – 2)(x – 10)?
I have
x2 – 12x + 20.
Who has
(x + 7)(x – 3)?
I have
x2 + 4x – 21.
Who has
(x + 9)(x + 6)?
I have
x2 + 15x + 54.
Who has
(x – 8)(x – 7)?
I have
x2 – 15x + 56.
Who has
(x + 8)(x + 5)?
I have
x2 + 13x + 40.
Who has
(x – 6)(x – 10)?
I have
x2 – 16x + 60.
Who has
(x + 6)(x + 5)?
I have
x2 + 11x + 30.
Who has
(x – 12)(x + 3)?
I have
x2 – 9x – 36.
Who has
(x – 1)(x – 1)?
I have
x2 – 2x + 1.
Who has
(x + 3)(x + 3)?
I have
x2 + 6x + 9.
Who has
(x – 9)(x + 3)?
I have
x2 – 6x – 27.
Who has
(x – 7)(x + 3)?
NUCC | Secondary II Math 56
Unit 4.4
I have
x2 – 4x – 21.
Who has
(x – 1)(x + 10)?
I have
x2 + 9x – 10.
Who has
(x – 6)(x – 5)?
I have
x2 – 11x + 30.
Who has
(x – 3)(x – 3)?
I have
x2 – 6x + 9.
Who has
(x + 15)(x – 4)?
I have
x2 + 11x – 60.
Who has
(x + 5)(x + 7)?
I have
x2 + 12x + 35.
Who has
(x + 9)(x – 3)?
I have
x2+ 6x – 27.
Who has
(x + 6)(x + 10)?
I have
x2 + 16x + 60.
Who has
(x – 11)(x + 2)?
I have
x2 – 9x – 22.
Who has
(x – 2)(x – 7)?
I have
x2 – 9x + 14.
Who has
(x + 5)(x – 9)?
I have
x2 – 4x – 45.
Who has
(x + 9)(x + 3)?
I have
x2 + 12x + 27.
Who has
(x – 9)(x – 9)?
NUCC | Secondary II Math 57
Unit 4.4
Special Products Bell Quiz
Name _______________________________________________
Period ___________
Factor each polynomial.
1. 𝒙𝟐 + 𝟒𝒙 + 𝟑
2. 𝒙𝟐 − 𝒙 − 𝟒𝟐
3. 𝒙𝟐 − 𝟐𝟒𝒙 + 𝟏𝟒𝟒
4. 𝟓𝒙𝟐 + 𝒙 − 𝟑
5. 𝟔𝒙𝟐 + 𝟏𝟏𝒙 − 𝟏𝟎
NUCC | Secondary II Math 58
Unit 4.5
4.5 INTRO TO COMPLEX NUMBERS
Teacher Notes
Time Frame:
Materials Needed:
Related Standards: N.CN.1 Know there is a complex number i such that 𝑖 2 = −1, and every
complex number has the form 𝑎 + 𝑏𝑖 with a and b real.
Launch
John and Betty’s Story
Read the story of John and Betty up to page 14 where they introduce i. Print out the John and
Betty Guided notes page for each student. http://mathforum.org/johnandbetty

Answer the following questions as the story is read. Have the students fill out the first section
as you go (Answers are in blue):
1. What kind of numbers do John and Betty start out working with? Whole numbers
2. With three people they had to create Fractions?
3. When they needed to find the length of the sides of the square what kind of number was
used? Square Roots
4. Why did they create i ? So that it could multiply itself to get a negative number.
5. How is i defined in the story? 𝑖 ∙ 𝑖 = −1 or 𝑖 2 = −1
NUCC | Secondary II Math 59
Unit 4.5
Explore

In pairs or small groups of students answer the following questions.
6. Why did they use i?
7. Is i imaginary, like a make believe friend?
8. Think of looking in a mirror. What do you see?
a. Is it real?
b. Is it imaginary?
c. What does the image see when it looks in the mirror?
d. Is it real?
9. Answer again, why did they use i?
10. How is i defined in the story? If 𝑖 2 = −1 , what does i equal? 𝑖 = √−1
11. Can you mix real numbers with imaginary numbers? How would you represent real
numbers mixed with imaginary numbers?
Examples: 6 + 4𝑖; 5 − 3𝑖
Discuss

As a class discuss the students’ responses to the questions and the concept of imaginary
numbers.
12. Can you mix real numbers with imaginary numbers? How would you represent real
numbers mixed with imaginary numbers?
Examples: 6 + 4𝑖; 5 − 3𝑖
13. These are called complex numbers. Complex numbers are written in the form 𝑎 + 𝑏𝑖,
where a is the _real number_ part and bi is the _imaginary number_ part. Also, a is a
_real _ number and the coefficient b is a _real _ number.
14. Properties of taking a square root of a negative number.
If r is a positive real number, then √−𝑟 = 𝑖 √𝑟
It follows that (𝑖√𝑟)2 = −𝑟

What are some real world applications to imaginary numbers?
http://www.lessonplanet.com/article/math/real-world-applications-to-imaginary-andcomplex-numbers.
NUCC | Secondary II Math 60
Unit 4.5
Ready, Set, Go!
Complex Numbers Worksheet
NUCC | Secondary II Math 61
Unit 4.5
Mathematics Content
Cluster Title: Perform arithmetic operations with complex numbers.
Standard N.CN.1: Know there is a complex number i such that i2 = -1, and every complex
number has the form a + bi with a and b real.
Concepts and Skills to Master
•
•
Understand that the set of complex numbers includes the set of all real numbers and the set of
imaginary numbers.
Express numbers in the form a + bi.
Critical Background Knowledge
•
Real number system and its subsets..
Academic Vocabulary
real numbers, complex numbers, imaginary numbers, I, a + bi
Suggested Instructional Strategies
• Use the definition of square root and the
Fundamental Theorem of Algebra to
show the need for √−1 . Consider this in
Skills:
• Write √−25 + √9 as a complex number
in the form of a + bi.
a historical context.
•
•
Solve x2 – a2 = 0 and x2 + a2 = 0 where a
is an integer.
Connect imaginary solutions to the
graphs of quadratic functions.
Some Useful Websites:
http://mathforum.org/johnandbetty
http://www.lessonplanet.com/article/math/real-world-applications-to-imaginary-and-complexnumbers
NUCC | Secondary II Math 62
Unit 4.5
Complex Numbers:
Name _______________________________________________
Period ___________
Ready, Set, Go!
Ready
Prerequisite problems:
A. Simplify the expression.
1. √64
2. √28
3. √3 ∙ √27
4. 4√36
B. Solve the equation
5. 𝑠 2 = 169
6. 4𝑝2 = 448
7. 7𝑟 2 − 10 = 25
8. 𝑥 2 = 84
Today’s BIG idea.
C. Simplify
7. √−36
8. − √−16
9. 11√−81
10. 16 − √−16
12. 3√−9
13. −2√−16
14. 44 − √−1
16. 𝑥 2 + 11 = 3
17. 3𝑥 2 − 7 = −31
18. 5𝑥 2 + 33 = 3
Set
11. ±√−49
D. Solve the equation
15. 𝑠 2 = −13
E. In each expression identify the real number part a and the imaginary number part
bi.
19. 4 − 7𝑖
20. 5 + 4𝑖
21. 15
22. 12𝑖
Go
F. Mixed Review
23.
State the polynomial in standard form in regard to x: 4𝑥 2 − 8𝑥 + 𝑥 3 𝑦 + 11𝑦 2
24.
Factor the polynomial completely:
25.
Factor the special product:
26.
Find the missing angle:
27.
Solve the proportion:
126
𝑘
3𝑥 2 − 2𝑥 − 5
𝑥 2 + 4𝑥 + 4
=
14
3
NUCC | Secondary II Math 63
Unit 4.5
Name _______________________________________________
Period ___________
The Story of John and Betty Guided Notes
Answer the following questions as the story is read.
1. What kind of numbers do John and Betty start out working with?
2. With three people they had to create ________?
3. When they needed to find the length of the sides of the square what kind of number was
used?
4. Why did they create i?
5. How is i defined in the story?
In pairs or small groups discuss and answer the following questions.
6. Why did they use i?
7. Is i imaginary, like a make believe friend?
8. Think of looking in a mirror. What do you see?
 Is it real?
 Is it imaginary?
 What does the image see when it looks in the mirror?
 Is it real?
9. Answer again, why did they use i ?
10. How is I defined in the story? If i2 = -1; what does i equal?
11. Can you mix real numbers with imaginary numbers? How would you represent real numbers
mixed with imaginary numbers?
As a class discuss the responses to the questions and the concept of imaginary numbers.
12. Can you mix real numbers with imaginary numbers? How would you represent real numbers
mixed with imaginary numbers? (Discuss what you discovered in question 11)
13. These are called ___________________. Complex numbers are written in the form _______
where a is the ________ part and bi is the ___________ part. Also, a is a ______ number
and the coefficient b is a ______ number.
14. Properties of taking a square root of a negative number.
NUCC | Secondary II Math 64
Unit 4.6
4.6 OPERATIONS WITH COMPLEX NUMBERS
Teacher Notes
Time Frame:
Materials Needed:
Related Standards: N.CN.2 Use the relation 𝑖 2 = −1 and the commutative, associative, and
distributive properties to add, subtract and multiply complex numbers.
Launch
List the different number systems. Natural numbers, Integers, Rational and Irrational, Real
numbers, Imaginary numbers, Complex numbers.
Can you come up with a way to visually organize all these different number systems?
(example of one way):
Complex Numbers (C)
Real Numbers
(R)
Imaginary
Numbers
(Im)
Could we combine two complex numbers together? What kind of number would you get? Come
up with three ways to combine complex numbers. Can you write some examples? Add, subtract,
and multiply.
Explore
Mix –N –Match This activity is done with adding and subtracting complex numbers. Print
activity on heavier paper and cut into problems.
NUCC | Secondary II Math 65
Unit 4.6
Instructions for the activity.
1. Each student is given a card with a problem
to complete.
2. Once the question is answered they mix
around the middle of the room until they find
a match, another student with the same
answer. Once their match is found they move
to the outside of the room to discuss.
3. Partners quiz and discuss with each other
about how they found the solution. Students
coach and praise each other.
4. Students are then given another card or can
pass cards so each student has a new problem
for another round.
Discuss



What are the steps you took to add or subtract the complex numbers? Simplify any
negative square roots. Commutative and associative properties to combine real number
with real number and imaginary number with imaginary number.
What kind of numbers did you see in the solutions? Real numbers, imaginary numbers,
complex numbers
What about multiplying complex numbers? Are there properties that we can review to
help with multiplying? Distributive property; FOILing
NUCC | Secondary II Math 66
Unit 4.6
Assign Ready, Set, Go! Worksheet
NUCC | Secondary II Math 67
Unit 4.6
Mathematics Content
Cluster Title: Perform arithmetic operations with complex numbers.
Standard N.CN.2: Use the relation i2 = -1 and the commutative, associative, and distributive
properties to add, subtract, and multiply complex numbers.
Concepts and Skills to Master

Add, subtract, and multiply complex numbers.
Critical Background Knowledge

Definition of i.

Combining like terms in polynomials. (II.1.A.APR.1)
Academic Vocabulary
Complex numbers, i
Suggested Instructional Strategies
 Relate operations with complex numbers to
familiar operations with numbers or
polynomial expressions.
Skills:
 Perform the following operations and
simplify the solutions.
(2 + 3i) + (5 – 7i)
(3 – 5i)(2 + 4i)
√3(√-6 + 4)
Some Useful Websites:
NUCC | Secondary II Math 68
Unit 4.6
Teacher’s Master of Mix –N –Match
𝟏. (𝟖 − 𝟔𝒊) + (𝟕 + 𝟒𝒊)
2.(𝟐 − 𝟑𝒊) − (𝟔 − 𝟓𝒊)
3.(𝟐𝟑 + 𝟒𝒊) − (𝟐 + 𝟓𝒊)
4.(𝟑 + 𝟒𝒊) + (𝟔 + 𝟕𝒊)
15 − 2𝑖
−4 + 2𝑖
21 − 𝑖
9 + 11𝑖
𝟓. (𝟏𝟐 + 𝟔𝒊) − (−𝟑 + 𝟒𝒊)
6.(−𝟒 + 𝒊) − (𝟑 − 𝒊)
7.(𝟏𝟓 − 𝟑𝒊) + (𝟔 + 𝟐𝒊)
8.(𝟏𝟎 + 𝟒𝒊) − (𝟏 − 𝟕𝒊)
15 − 2𝑖
−4 + 2𝑖
21 − 𝑖
9 + 11𝑖
9.(𝟏𝟔𝟑𝒊) + (𝟒 + 𝟐𝒊)
10.(𝟏𝟖 + 𝟕𝒊) + (−𝟑 + 𝟏𝟔𝒊)
11.(−𝟏𝟐 − 𝟒𝒊) + (−𝟏𝟎 − 𝟑𝒊)
12.(−𝟖 + 𝟑𝒊) + (−𝟕 − 𝟐𝒊)
20 − 𝑖
15 + 23𝑖
−22 − 7𝑖
−15 + 𝑖
13.(𝟏𝟐 + 𝟔𝒊) + (𝟖 − 𝟕𝒊)
14.(𝟓 + 𝟏𝟏𝒊) + (𝟏𝟎 + 𝟏𝟐𝒊)
15.(−𝟏𝟏 − 𝟑𝒊) − (𝟏𝟏 + 𝒊)
16.(𝟑 + 𝟗𝒊) − (𝟏𝟖 + 𝟖𝒊)
20 − 𝑖
15 + 23𝑖
−22 − 7𝑖
−15 + 𝑖
18.(−𝟐 + 𝟓𝒊) + (𝟐 − 𝟏𝟓𝒊)
19.(𝟏𝟒 + 𝟐𝟔𝒊) − (𝟕 + 𝟑𝒊)
20.(𝟐𝟒 + 𝟏𝟔𝒊) − (𝟏𝟓 + 𝟒𝒊)
0
7 + 23𝑖
9 + 12𝑖
17.(−𝟔𝟑 − 𝟏𝟕𝒊) − (𝟐𝟓 − 𝟕𝒊)
−19
21.(𝟔 − 𝟕𝒊) − (𝟐𝟓 − 𝟕𝒊)
−19
25.(−𝟏𝟒𝟒 + 𝟏𝟐𝒊) − (𝟐𝟒 + 𝟏𝟔𝒊)
−120 − 4𝑖
29.(−𝟔𝟎 − 𝟏𝟎𝒊) − (𝟔𝟎 − 𝟔𝒊)
−120 − 4𝑖
33.(𝟖 − 𝟏𝟓𝒊) − (𝟏𝟎 − 𝟑𝒊)
−2 − 12𝑖
37.(𝟔 − 𝟔𝒊) − (𝟖 + 𝟔𝒊)
−2 − 12𝑖
22.(−𝟏𝟐 + 𝟒𝒊) − (−𝟏𝟐 + 𝟒𝒊)
0
26.(𝟏𝟒 − 𝟑𝒊) − (𝟐𝟎 + 𝟐𝒊)
−6 − 5𝑖
30.(−𝟒 + 𝟐𝒊) + (−𝟐 − 𝟕𝒊)
−6 − 5𝑖
34.(𝟑 + 𝟗𝒊) − (𝟒 + 𝟐𝒊)
−1 + 7𝑖
38.(𝟗 + 𝟏𝟏𝒊) − (𝟏𝟎 + 𝟒𝒊)
−1 + 7𝑖
23.(𝟐 + 𝟏𝟒𝒊) − (−𝟓 − 𝟗𝒊)
7 + 23𝑖
9 + 12𝑖
27.(−𝟐𝟒 − 𝟔𝒊) − (−𝟐𝟖 + 𝟔𝒊)
4 − 12𝑖
28.(𝟐 + 𝟏𝟓𝒊) + (𝟏𝟖 + 𝟒𝒊)
20 + 19𝑖
31.(𝟗 − 𝟖𝒊) − (𝟓 + 𝟒𝒊)
4 − 12𝑖
32.(𝟏𝟔 + 𝟕𝒊) + (𝟒 + 𝟏𝟐𝒊)
20 + 19𝑖
35.(𝟏𝟕 − 𝟐𝒊) + (−𝟓 + 𝟕𝒊)
12 + 5𝑖
36.(𝟔 − 𝟓𝒊) + (𝟒 + 𝟐𝒊)
10 − 3𝑖
39.(𝟕 + 𝒊) + (𝟓 + 𝟒𝒊)
12 + 5𝑖
24.(𝟔 + 𝟏𝟒𝒊) − (−𝟑 + 𝟐𝒊)
40.(𝟏𝟕 − 𝟏𝟐𝒊) − (𝟕 + 𝟏𝟓𝒊)
10 − 3𝑖
NUCC | Secondary II Math 69
Unit 4.6
Student copy of Mix –N –Match
𝟏. (𝟖 − 𝟔𝒊) + (𝟕 + 𝟒𝒊)
2.(𝟐 − 𝟑𝒊) − (𝟔 − 𝟓𝒊)
3.(𝟐𝟑 + 𝟒𝒊) − (𝟐 + 𝟓𝒊)
4.(𝟑 + 𝟒𝒊) + (𝟔 + 𝟕𝒊)
𝟓. (𝟏𝟐 + 𝟔𝒊) − (−𝟑 + 𝟒𝒊)
6.(−𝟒 + 𝒊) − (𝟑 − 𝒊)
7.(𝟏𝟓 − 𝟑𝒊) + (𝟔 + 𝟐𝒊)
8.(𝟏𝟎 + 𝟒𝒊) − (𝟏 − 𝟕𝒊)
9.(𝟏𝟔𝟑𝒊) + (𝟒 + 𝟐𝒊)
10.(𝟏𝟖 + 𝟕𝒊) + (−𝟑 + 𝟏𝟔𝒊)
11.(−𝟏𝟐 − 𝟒𝒊) + (−𝟏𝟎 − 𝟑𝒊)
12.(−𝟖 + 𝟑𝒊) + (−𝟕 − 𝟐𝒊)
13.(𝟏𝟐 + 𝟔𝒊) + (𝟖 − 𝟕𝒊)
14.(𝟓 + 𝟏𝟏𝒊) + (𝟏𝟎 + 𝟏𝟐𝒊)
15.(−𝟏𝟏 − 𝟑𝒊) − (𝟏𝟏 + 𝒊)
16.(𝟑 + 𝟗𝒊) − (𝟏𝟖 + 𝟖𝒊)
17.(−𝟔𝟑 − 𝟏𝟕𝒊) − (𝟐𝟓 − 𝟕𝒊)
18.(−𝟐 + 𝟓𝒊) + (𝟐 − 𝟏𝟓𝒊)
19.(𝟏𝟒 + 𝟐𝟔𝒊) − (𝟕 + 𝟑𝒊)
20.(𝟐𝟒 + 𝟏𝟔𝒊) − (𝟏𝟓 + 𝟒𝒊)
21.(𝟔 − 𝟕𝒊) − (𝟐𝟓 − 𝟕𝒊)
22.(−𝟏𝟐 + 𝟒𝒊) − (−𝟏𝟐 + 𝟒𝒊)
23.(𝟐 + 𝟏𝟒𝒊) − (−𝟓 − 𝟗𝒊)
24.(𝟔 + 𝟏𝟒𝒊) − (−𝟑 + 𝟐𝒊)
25.(−𝟏𝟒𝟒 + 𝟏𝟐𝒊) − (𝟐𝟒 + 𝟏𝟔𝒊)
26.(𝟏𝟒 − 𝟑𝒊) − (𝟐𝟎 + 𝟐𝒊)
27.(−𝟐𝟒 − 𝟔𝒊) − (−𝟐𝟖 + 𝟔𝒊)
28.(𝟐 + 𝟏𝟓𝒊) + (𝟏𝟖 + 𝟒𝒊)
29.(−𝟔𝟎 − 𝟏𝟎𝒊) − (𝟔𝟎 − 𝟔𝒊)
30.(−𝟒 + 𝟐𝒊) + (−𝟐 − 𝟕𝒊)
31.(𝟗 − 𝟖𝒊) − (𝟓 + 𝟒𝒊)
32.(𝟏𝟔 + 𝟕𝒊) + (𝟒 + 𝟏𝟐𝒊)
33.(𝟖 − 𝟏𝟓𝒊) − (𝟏𝟎 − 𝟑𝒊)
34.(𝟑 + 𝟗𝒊) − (𝟒 + 𝟐𝒊)
35.(𝟏𝟕 − 𝟐𝒊) + (−𝟓 + 𝟕𝒊)
36.(𝟔 − 𝟓𝒊) + (𝟒 + 𝟐𝒊)
37.(𝟔 − 𝟔𝒊) − (𝟖 + 𝟔𝒊)
38.(𝟗 + 𝟏𝟏𝒊) − (𝟏𝟎 + 𝟒𝒊)
39.(𝟕 + 𝒊) + (𝟓 + 𝟒𝒊)
40.(𝟏𝟕 − 𝟏𝟐𝒊) − (𝟕 + 𝟏𝟓𝒊)
NUCC | Secondary II Math 70
Unit 4.6
Name _______________________________________________
Period ___________
Ready, Set, Go!
Ready
Prerequisite problems: (commutative, associative, distributive properties)
A.
Identify the property that the statement illustrates.
1. (4 + 9) + 3 = 4 + (9 + 3)
B.
2. 7(2 + 8) = 7(2) + 7(8)
3.(12𝑏 + 15) − 3𝑏 = 15 + 9𝑏
Find the product.
4. 𝑖 ∙ 𝑖
5. (𝑥 + 6)(𝑥 + 3)
6. (𝑥 − 5)2
7. 4(𝑥 + 5)(𝑥 − 5)
Set
Today’s BIG idea.
C.
Write the expression as a complex number in standard form.
8. (6 − 3𝑖) + (5 + 4𝑖)
9. (9 + 8𝑖) + (8 − 9𝑖)
10. (−2 − 6𝑖) − (4 − 6𝑖)
11. (−1 + 𝑖) − (7 − 5𝑖)
12. (8 + 20𝑖) − (−8 + 12𝑖)
13. (−1 + 4𝑖) + (−9 − 2𝑖)
14. 6𝑖(3 + 2𝑖)
15. −𝑖(4 − 8𝑖)
16. (−2 + 5𝑖)(−1 + 4𝑖)
17. (−1 − 5𝑖)(−1 + 5𝑖)
18. (8 − 3𝑖)(8 + 3𝑖)
Go
Challenge
D.
Write the expression as a complex number in standard form.
19. −8 − (3 + 2𝑖) − (9 − 4𝑖)
20. (3 + 2𝑖) + (5 − 𝑖) + 6𝑖
21. 5𝑖(3 + 2𝑖)(8 + 3𝑖)
22. (1 − 9𝑖)(1 − 4𝑖)(4 − 3𝑖)
NUCC | Secondary II Math 71
Unit H4.7
H4.7
FACTORING WITH COMPLEX
NUMBERS
Teacher Notes
Time Frame:
Materials Needed:
Related Standards: N.CN.H.8 Extend polynomial identities to the complex numbers.
For example, rewrite 𝑥 2 + 4 as (𝑥 + 2𝑖)(𝑥 − 2𝑖).
Launch
What are some polynomial identities? When we multiply or factor a polynomial, we see the same
patterns over and over again.
As learned in 4.4:
Difference of Two Squares
Perfect Square Sums
Perfect Square Differences
Sum of Two Cubes
Difference of Two Cubes
These same patterns are also true when we deal with complex numbers. What pattern helped us
to factor a different of squares? For example 𝑥 2 − 9
𝑥 2 − 9 = (𝑥 + 3)(𝑥 − 3)
What if we try to factor 𝑥 2 + 9?
Explore
First consider multiplying these two complex numbers: (𝑥 + 𝑖)(𝑥 − 𝑖)
Remember that 𝑖 ∙ 𝑖 = −1 and use the FOIL method to multiply.
(𝑥 + 𝑖)(𝑥 − 𝑖) = 𝑥 2 − 𝑥𝑖 + 𝑥𝑖 − 𝑖 2 = 𝑥 2 − (−1) = 𝑥 2 + 1
Now try (𝑥 + 3𝑖)(𝑥 − 3𝑖).
(𝑥 + 3𝑖)(𝑥 − 3𝑖) = 𝑥 2 − 3𝑥𝑖 + 3𝑥𝑖 − 9𝑖 2 = 𝑥 2 − 9(−1) = 𝑥 2 + 9
What pattern are you seeing?
Based on this pattern could you factor other Sum of two squares polynomials?
How would you factor 𝑥 2 + 25? Check your answer.
𝑥 2 + 25 = (𝑥 + 5𝑖)(𝑥 − 5𝑖)
Check: (𝑥 + 5𝑖)(𝑥 − 5𝑖) = 𝑥 2 + 5𝑥𝑖 − 5𝑥𝑖 − 25𝑖 2 = 𝑥 2 − 25(−1) = 𝑥 2 + 25
Try 9𝑥 2 + 64. Is this a sum of two squares? Yes (3𝑥)2 + 82
NUCC | Secondary II Math 72
Unit H4.7
What is the imaginary part going to be for each factor? 8𝑖
What are the factors for this sum of two squares?
(3𝑥 + 8𝑖)(3𝑥 − 8𝑖)
Discuss
Remember it is not possible to factor the sum of two squares over real numbers. You can factor
the difference of two squares over real numbers. The sum of two squares must be factored over
complex numbers.
Assign Ready, Set, Go!
NUCC | Secondary II Math 73
Unit H4.7
Mathematics Content
Cluster Title: Use complex numbers in polynomial identities and equations.
Standard N.CN.H.8: Extend polynomial identities to the complex number. (For example,
rewrite 𝑥 2 + 4 as (𝑥 + 2𝑖)(𝑥 − 2𝑖).)
Concepts and Skills to Master

Express a quadratic as a product of two complex factors.
Critical Background Knowledge





Factor quadratics.
Understand that some quadratic functions have complex solutions.
Know the definition of i.
Perform operations on complex numbers.
Standard form of a complex number.
Academic Vocabulary
conjugates, complex numbers, i, factor
Suggested Instructional Strategies
Skills:
 Demonstrate that any binomial quadratic
 Factor over the complex number system.
expression can be expressed as the
x2 - 16
Answer: (x + 4i)(x-4i)
2
2
2
difference of two squares (e.g., x + 16 = x
x -10x + 34
Answer (x + 5i)(x-5i)
– 16i2).
Some Useful Websites:
NUCC | Secondary II Math 74
Unit H4.7
Name _______________________________________________
Period ___________
Ready, Set, Go!
Ready
Prerequisites
A. Factor the expression
1. 𝑥 2 − 36
2. 4𝑏 2 − 81
3. 36𝑛2 − 9
4. 5𝑥 2 − 45
5. 𝑥 3 − 3𝑥 2 − 16𝑥 + 48
6. 2𝑥 3 − 7𝑥 2 − 8𝑥 + 28
Set
B. Factor the polynomial completely
7. 𝑥 2 + 36
8. 𝑥 2 + 100
9. 𝑥 4 − 16
10. 𝑥 3 + 4𝑥
11. 25𝑥 2 + 49
12. 16𝑥 2 − 25
13. 121𝑥 2 − 36𝑦 2
14. 𝑥 3 + 2𝑥 2 + 𝑥 + 2
15. 𝑥 3 − 2𝑥 2 + 16𝑥 − 32
Go
Mixed Review
C. Simplify each expression
16. ±√−49
17. −√−16
18. 11√−81
D. Find the error in the student’s work
19. 14 − √−16
14 + √16
14 + 4
18
NUCC | Secondary II Math 75
Unit 4
Practice Exam Secondary II Unit 4
Name _______________________ Hour _____
1. Write the polynomial in standard form and state the degree. 4𝑥 3 − 3𝑥 2 + 𝑥 5 + 5𝑥 7 − 9
2. Write an English statement for the algebraic expression. (3𝑥)2 + 𝑥
Simplify the expression
3. 3𝑦 2 + 5𝑥 − 12𝑥 + 9𝑦 2 − 5
4. 2(𝑥 + 4)(𝑥 − 1)
5. (5𝑥 2 − 7𝑥 + 2) ÷ (𝑥 − 1)
Factor the expression completely
6. 𝑎2 − 13𝑎 + 22
7. 12𝑥 2 − 4𝑥 − 40
8. 12𝑚2 − 36𝑚 + 27
Factor the special product expression
9. 4𝑟 2 − 25
10. 25𝑡 2 − 30𝑡 + 9
NUCC | Secondary II Math 76
Unit 4
11. 32𝑣 2 − 2
12. 27𝑚3 + 1
13. −5𝑧 3 + 320
Write the expression as a complex number in standard form
14. (−2 − 6𝑖) − (4 − √−36)
15. (8 − 5𝑖) − (−11 + 4𝑖)
16. 6𝑖(3 + 2𝑖)
17. (5 − 7𝑖)(−1 − 3𝑖)
Factor the polynomial completely
18. 4𝑥 2 + 25
19. 3𝑥 2 + 48
20. 𝑥 3 + 2𝑥 2 + 4𝑥 + 8
NUCC | Secondary II Math 77
Unit 4
Exam Secondary II Unit 4
Name _____________________Hour________
1. Write the polynomial in standard form and state the degree. 𝑥 6 + 6𝑥 2 − 4𝑥 3 + 7
2. Write and English statement for the algebraic expression.
2𝑦 2
3
+𝑦
Simplify the expression
3. 7𝑥 2 + 8 − 3𝑥 − 5𝑥 2
4. 𝑥(2𝑥 + 1)(𝑥 − 3)
5. (4𝑥 2 − 12𝑥 + 5) ÷ (2𝑥 − 1)
Factor the expression completely
6. 𝑥 2 + 8𝑥 − 65
7. 20𝑥 2 + 124𝑥 + 24
8. −8𝑦 2 + 28𝑦 − 60
Factor the special product expression
9. 49𝑥 2 − 16
10. 12𝑚2 − 36𝑚 + 27
11. 36𝑥 2 − 84𝑥 + 49
NUCC | Secondary II Math 78
Unit 4
12. 27𝑎3 − 1000
13. 8𝑐 3 + 343
Write the expression as a complex number in standard form.
14. (−1 + √−1) − (7 − √−25)
15. (14 + 3𝑖) + (7 + 6𝑖)
16. −𝑖(4 − 8𝑖)
17. (5 − 7𝑖)(−4 − 3𝑖)
Factor the polynomial completely
18. 16𝑥 2 + 49
19. 8𝑥 2 + 50
20. 9𝑥 3 + 27𝑥 2 + 16𝑥 + 48
NUCC | Secondary II Math 79