POLYNOMIALS and FACTORING
Exponents (2 days);
Overview of Objectives, students should be able to:
1.
Evaluate exponential expressions
2.
Use the product rule for exponents,
3.
Use the power rule for exponents,
4.
Use the power rules for products and quotients,
5.
Use the quotient rule for exponents, and define a number
raised to the 0 power
Main Overarching Questions:
1. How do you remember the rules for exponents?
2. How do you decide which rule to use when simplifying expressions involving exponents?
6. Be able to decide which rule to use
Objectives:
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Activities and Questions to ask students:
Evaluate exponential expressions
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Begin with simple exponents: e.g . 32 ,33 , 23 , 24 , and ask students to evaluate.
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3
3
4
What happens when the negative is there? e.g . − 3 , −2 , −2
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m+n
Use the product rule for exponents, a ⋅ a =
a
m
n
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What happens when you have ( −3) , ( −2 ) , ( −2 )
3
3
4
?
What is the difference between the two examples?
2
3
3
4
Then look at what happens when you multiply the same bases: e.g . 3 ⋅ 3 , 2 ⋅ 2
Have students work through several examples asking whether they see a pattern. Does a
pattern exist?
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3
4
7
Have students work through that 2 ⋅ 2 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 2
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2
3
3
4
How about e.g . a ⋅ a , b ⋅ b ?
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What pattern do you notice and how can you represent that pattern?
m
n
a m+n
Have students draw to the conclusion that a ⋅ a =
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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( )
Use the power rule for exponents, a
m
n
= a mn
Use the power rules for products ( ab ) = a b
n
n n
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3
Now try examples involving powers: e.g .
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What does the 4 mean in 32
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How is this similar to 32 ⋅ 33 example from before?
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Does a pattern exist?
( )
2
3
4
4
Have students work through several examples asking whether they see a pattern.
( ) , (b ) ?
3
4
•
2
How about a
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What pattern do you notice and how can you represent that pattern?
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m
Have students draw to the conclusion that a
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Now try examples involving powers: e.g .
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What if you have ( ab ) , ( xy ) , or ( 3 x ) instead?
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Students do several examples asking whether they see a pattern.
Can we use our other rules to help us?
Does a pattern exist?
What pattern do you notice and how can you represent that pattern?
3
( )
3
4
n
= a mn
(3 ⋅ 2) , ( 4 ⋅ 2)
3
2
2
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Have students draw to the conclusion that ( ab ) = a n b n
•
2
Now try examples involving powers: e.g .  
3
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Do several examples asking students whether they see a pattern.
Does a pattern exist?
•
a
Ask them whether it would be try for  
b
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What pattern do you notice and can you represent that pattern?
•
an
a
Have students draw to the conclusion that   = n
b
b
n
an
a
Use the power rules for quotients,   = n
b
b
(3 ) , ( 2 )
•
n
3

,

3
4
x
 ?
y
n
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Use the quotient rule for exponents,
am
= a m − n , and
an
define a number raised to the 0 power
•
•
Begin with the quotient rule, but use the same base.
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Does a pattern exist?
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Be able to decide which rule to use
Do several examples asking students whether they see a pattern.
What pattern do you notice and can you represent that pattern?
am
m−n
Have students draw to the conclusion that n = a
a
Can you define what it means to be raised to the 0 power? Help students define.
(x
) ( )
3
⋅ y 3 , 3z 3
2
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Begin with an example like:
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Is there a strategy for picking which rule to use?
Explain how you would decide which rule to use?
Give worksheet on more difficult problems. Students can work in groups
Ask students to talk through their strategy on how to solve.
Ask students if they performed them in a different way?
Is it correct to say there are many ways to simplify? Why and explain?
2
Negative Exponents and Scientific Notation (2 days);
Overview of Objectives, students should be able to:
1.
Simplify expressions containing negative exponents
2.
Use the rules and definitions for exponents to simplify
exponential expressions
3.
Write numbers in scientific notation
4.
Convert numbers in scientific notation to standard form
Objectives:
Main Overarching Questions:
1. What does it mean to have a negative exponent?
2. How can you represent very small or very large numbers using exponents?
Activities and Questions to ask students:
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Simplify expressions containing negative exponents
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•
•
•
am
= a m−n
n
a
Look at the rule:
What happens to the expression when n is greater than m?
What does the fraction look like? Where does the base end up?
What happens when the negative exponent is in the denominator?
Use the rules and definitions for exponents to simplify
exponential expressions
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Handout of worksheet for students to simplify expressions that also involve negative
exponents.
When complete ask students to explain how they simplified.
Ask if anyone did it differently and ask them to explain their method.
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Write numbers in scientific notation
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Can you represent very large numbers using exponents? 2,100,000,000,000,000,000.
Start with 310. Is this equal to 3.1 x 10? Then use 3.1 x 101
Can you represent 3100 by 3.1 x 100? Then use 3.1 x 102
How would you present 31000 using this pattern?
Then try 3,100,000,000.
What number would 4.134 x 106 be?
Now look at 0.44. Is this equal to 4.4 x 10-1?
Can you represent 0.0044 using this type of pattern?
Is there a general strategy for writing small or large numbers using exponents?
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Convert numbers in scientific notation to standard form
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Can we extend strategies learned with exponents and scientific notation to simplify these
expressions? (2 x 106)(3 x 10-3)
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How about expressions like these? How would you simplify these expressions?
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8x1010
4x103
Introduction to Polynomials (1 day)
Overview of Objectives, students should be able to:
1.
Define term and coefficient of a term
2.
Define polynomial, monomial, binomial, trinomial, and
Main Overarching Questions:
1.
What is the difference between a monomial, binomial, trinomial, etc.?
2. What is a like term?
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U.S. Department of Education, and you should not assume endorsement by the Federal Government.
degree
3. Why would you want to simplify a polynomial?
3.
Evaluate polynomials for given replacement values
4. When do you simplify a polynomial?
4.
Simplify a polynomial by combining like terms
5. How do you know whether a polynomial is considered simplified?
5.
Simplify a polynomial in several variables
Objectives:
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Define term and coefficient of a term
Define polynomial, monomial, binomial, trinomial, and
degree
Activities and Questions to ask students:
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Evaluate polynomials for given replacement values
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Provide students the definitions for term and coefficient. Some examples should contain one
variable only and others with several variables.
Give students several examples to talk through.
What is a constant term? What does the word constant mean?
What is a numerical coefficient?
What can these terms represent?
Ask students to list words having the prefixes mono-, bi-, tri-, and poly-.
Allow students to discuss the meanings of the prefixes based on the words on their lists.
Provide several examples containing a monomial, binomial, trinomial, and higher degree
polynomials.
Ask students to apply the meanings of the prefixes to guess the definitions of monomial,
binomial, trinomial, and polynomial.
What difference do you notice with these examples?
What is a polynomial?
If we call the two term polynomial a Binomial, what do you think we call a three term
polynomial?
How do you know what is the degree of the polynomial?
How do you evaluate a polynomial given replacement values?
Have students work in groups to evaluate polynomials.
Give several word problems like “Finding Free-fall Time” from a building. A building is 466.3
feet tall. An object is dropped from the highest point. Neglecting air resistance, the height in
feet of the object above ground at time t seconds is given by -16t2+466.3. Find the height of
the object when t = 1 second, and when t = 3 seconds, etc.
What do these answers mean?
Ask students to think about how the given equation relates to the problem.
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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Simplify a polynomial by combining like terms
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Simplify a polynomial in several variables
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What does 466.3 represent in the equation?
What if the building was 632.7 feet tall? How would the equation differ?
Do you think this polynomial will give a good estimate of the height of the object for all
values of t? --They should say no, because once the object hits the ground the polynomial
does not apply.
What is a “like term”?
Give several examples: {5x2, -7x2}, {a2b3, -3ab3} , etc.
Which of those are “like terms”?
How do you think you can simplify terms that are alike?
What if you have 3 squares + 4 squares? Do you have 7 squares?
Can you combine 7 squares + 3 watermelons?
What are the like terms?
Recall that like terms may have several variables.
Have students work in groups to combine several polynomials involving single variables and
multiple variables.
What did everyone get for each answer and have student discuss their findings.
Adding and Subtracting Polynomials (1 day)
Overview of Objectives, students should be able to:
1.
Add polynomials
2.
Subtract polynomials
3.
Add or subtract polynomials in one variable
4.
Add or subtract polynomials in several variables
Objectives:
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Add polynomials
Main Overarching Questions:
1.
How do you add and/or subtract polynomials?
2.
How do you know you are finished?
Activities and Questions to ask students:
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Recall what it means to be “like” terms.
Look at several examples of addition. For example (3x5 – 7x3 + 2x – 1) + (3x3 -2x)
Ask students to combine like terms.
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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Subtract polynomials
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Add or subtract polynomials in one variable
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Add or subtract polynomials in several variables
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Have them discuss their answers. If they have differences, have them work out which is
correct and why.
What do you think happens when you subtract? (3x5 – 7x3 + 2x – 1) - (3x3 -2x)
Think about 75 – (6+2). Are you subtracting 8 or just 6? So, you subtract the entire
parenthesis? 75 – 6 -2
How about 75 – (8 – 3)? Are you subtracting out 8? 3? 5? Which one? What do we do if we
introduce variables?
Have students draw to the conclusion that the minus sign must be distributed.
What do we mean when we are subtracting polynomials?
Pose this question to the group and have the groups create the problem and simplify:
“Subtract the sum of (3x+6) and (8x-5) from (5x+2)”
Have students provide their answer. If there are discrepancies, have students determine
which is correct.
Is there a method to which you simplify these expressions?
Based upon what we’ve learned, is there going to be any difference in how you simplify
problems with several variables? If any, what is that difference?
(2a2 – ab + 6b2) + (-3a2 + ab – 7b2)
(5x2y2 + 3 – 9x2y + y2) – ( -x2y2 + 7 – 8xy2 + 2y2)
Multiplying Polynomials (1 day)
Overview of Objectives, students should be able to:
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Multiply monomials
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Multiply a monomial by a polynomial
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Multiply two polynomials
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Multiply polynomials vertically
Objectives:
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Multiply monomials
Main Overarching Questions:
1. How do you multiply polynomials?
2.
Why do you FOIL when you have two binomials?
Activities and Questions to ask students:
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Recall how to multiply monomials with the same base: x * x3 = x4 or 6a2*7a3b2 = 42a5b2
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Multiply a monomial by a polynomial
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Multiply two polynomials
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Multiply polynomials vertically and/or the box method.
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How would you multiply 7(6+3+2)? What is the answer? Did you multiply 7 to the entire
summation or to 6 only? Have students come to a consensus that you multiply 7 to the
entire summation.
How would you apply this same principle to 7(x+2), 7x(x+2), 7x(3x2 + x + 2)?
Have students come to the conclusion that you multiply the monomial to each term inside
the parenthesis.
Always remember to simplify as much as possible when you’ve multiplied.
For example: 3x2(2x+3) + x (2x - 5) = 6x3+9x2+2x2-5x
Are we done yet? No. Can you simplify? Yes, it equals 6x3+11x2-5x
Recall when multiplying, we had to multiply the monomial to EVERY term inside the
parenthesis.
What do you think happens when we have a binomial multiplied to another binomial?
(x+3)(x-2)
Have students talk about what they think you should do to multiply this out.
Have them come to the conclusion that you should multiply each term inside one parenthesis
to each term in the next parenthesis.
How would it change if we had two variables in there? (x+3y)(x-2y)
Have students come to the conclusion that it does not change the way they multiply.
Show students the box method: (3x+2)(2x-5)
3x
2
2
2x
6x
4x
-5
-15x
-10
Show students the vertical method. Show them the same example so they see the result is
the same.
Give a worksheet with several examples to work with in groups. Have them try all three
methods: box, vertical, and distributive methods.
In whole class discussion, have students talk about which method they prefer and why.
Special Products (3 days)
Overview of Objectives, students should be able to:
1.
Multiply two binomials using the FOIL method
Main Overarching Questions:
1. Why don’t you just square the two terms when you have a square of a binomial?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
2.
Square of a binomial
3.
Multiply the sum and difference of two terms
4.
Use special products to multiply binomials
Objectives:
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Multiply two binomials using the FOIL method
2.
How do you recognize these special products?
Activities and Questions to ask students:
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Square of a binomial
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Multiply the sum and difference of two terms
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We’ve seen the box method, vertical method and the distributive property method, let’s look
at the FOIL method. Once you become familiar with all four, you can use which ever method
you prefer.
Point out that special products are shortcuts for multiplying binomials – they can STILL be
worked out using the methods seen earlier.
Have student multiply (4x+3)2. Then have them multiply (4x-3)2. What is the relationship
between the problem and its solution? Do you recognize a pattern?
Return to the box method: (a+b)(a+b)
a
b
2
a
a
ab
b
ab
b2
Can you recognize a pattern?
How about this: (x+3)2
x
3
x
x2
3x
3
3x
32
How about this: (3x+y)(3x+y)
3x
y
3x
(3x)2
3xy
y
3xy
y2
Can you recognize a pattern?
Have students come to the conclusion that (a+b)2 = a2 +2ab +b2
Have student multiply (4x-3) (4x+3). What about (x+y)(x-y)? What is the relationship
between the problem and its solution? Do you recognize a pattern?
Return to the box method: (a+b)(a-b) = a2 – b2
a
b
2
a
a
ab
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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Use special products to multiply binomials
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-b
-ab
Can you recognize a pattern?
How about this: (x+3)(x-3) = x2 - 9
x
x
x2
-3
-3x
-b2
3
3x
-32
How about this: (3x+y)(3x-y) = 9x2 – y2
3x
y
3x
(3x)2
3xy
-y
-3xy
-y2
Can you recognize a pattern?
Have students come to the conclusion that (a+b)(a-b) = a2 - b2
Give students worksheets to work in groups.
Have students report their findings/answers. If there are any differences in answers, have
student discuss which is correct.
Dividing Polynomials (2 days)
Overview of Objectives, students should be able to:
1.
2.
Divide a polynomial by a monomial,
a+b a b
=
+ , where c ≠ 0
c
c c
1. Why do you need a common factor in each term to cross out?
2.
What are the similarities between using long division in arithmetic versus variables?
Use long division by a polynomial other than a monomial
Objectives:
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Main Overarching Questions:
Divide a polynomial by a monomial
Activities and Questions to ask students:
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Can you divide
x+3
0
? Why or why not? What about
? What is the difference
0
x+3
between the two examples? Why can you divide one of them to get 0, but not the other?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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How do you remember which is which? Have students explain their rationale.
Let’s look at an example where the denominator is not equal to 0. How can you simplify
3x + 6
? Have students guess. If they guess to just cross out the 3, to equal x+6, ask them is
3
3⋅ 2 + 6 3 ⋅ 2 + 6
=
= 2 + 6 = 8? Why or why not? Have students come to the conclusion
3
3
3x + 6
that you must cross out factor of 3 from EVERY term.
= x+2
3
Check their answer by plugging in x = 1, 2, and 3, into the original expression as well as the
resultant to make sure they match.
Point out that simplifying is not CHANGING the result – it just makes it look prettier and
SIMPLIER
Have students work through several examples in pairs and discuss the results. Have students
come to a general consensus on the correct answers.
The Greatest Common Factor (1 day)
Overview of Objectives, students should be able to:
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Find the GCF of a list of numbers
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Find the GCF of a list of terms
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Factor out the GCF from the terms of a polynomial
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Factor by grouping
Objectives:
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Find the GCF of a list of numbers
Main Overarching Questions:
1. Why do you need to find the GCF?
2. How can you check to see if you factored correctly?
3. Is it factored completely?
Activities and Questions to ask students:
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Point out to students that they already have the tools to solve linear equations such as 8x = x12, but have not learned the tools to involve equations in involving higher order polynomials
such as x2+8x = x-12. The factoring skills students learn in the remaining sections will give
them the tools needed to begin solving these more complicated kinds of equations.
Recall multiplying out 5(x+2) = 5x+10. This is multiplying.
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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Find the GCF of a list of terms
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Factor out the GCF from the terms of a polynomial
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Factor by grouping
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NOW, we will look at 5x + 10 = 5(x+2). This is factoring. (Draw a diagram to illustrate the
reverse nature)
BUT FIRST, we have to look at something called the Greatest Common Factor.
In the product 3*5 = 15, 3 and 5 are called factors and 3*5 is called the factored form of 15.
What are the factors of 4, 14, 28? What do these numbers have in common?
What do you think GREATEST Common Factor means? What would be the GCF of 4, 14, 28?
What would be the GCF of 4 and 28?
How about 60 and 24?
Let’s extend this to variables. What does x3 mean? Have students conclude that it is x*x*x.
“x” is a factor and there are 3 of them.
Ask students to talk about how they would find the GCF for y4 and y6.
What is the GCF of 1 and 3x?
What is the GCF of 6x2, 9x4, and -12x5?
Have students work through a worksheet with 5 different examples and discuss their results.
Have all the students come to a consensus on the correct answers.
From the example above, how can you factor 6x2+9x4 -12x5 knowing the GCF?
Factor -10x3 + 8x2 - 2x. Point out that we don’t like a (-) sign in the front of a trinomial. How
would you factor this polynomial?
Give students worksheet for group work. Have all the students come to a consensus on the
correct answers.
Ask students to construct a trinomial whose GCF is 4y3. Have students discuss their
trinomials and ask them if 4y3 is indeed the GCF. Point out that there are many answers to
this.
Give students an example like 2xy + 5y2 – 4x – 10y. Ask if they can see a GCF for this
example. Have them talk about the ways you could group terms together to form GCF for
those groups. When they are about to find the following: y(2x + 5y) – 2(2x + 5y), ask what do
you notice?
Can you factor that some more? Why or why not? What would it become? (y-2)(2x+5y)
Try another example 16x3 – 28x2 - 12x – 21. Answer: (4x-7)(4x2+3)
This is called Factoring by Grouping --- since you ‘grouped’ terms together that have GCF
Explore those answers where students group incorrectly!! You must have a common factor
to be able to factor completely.
Give a worksheet for students to work on is groups. Discuss results for a consensus on the
answers.
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
Factoring Trinomials of the Form x 2 + bx + c (1 day)
Overview of Objectives, students should be able to:
1.
Factor trinomials of the form x + bx + c by unFOILing
2.
Factor out the GCF and then factor trinomials of the form
Main Overarching Questions:
1. What does it mean to factor?
2
x 2 + bx + c
2. What is the end result when you factor?
3. What is the difference between multiplying polynomials and factoring polynomials?
4. How can you check to see if you factored correctly?
5. Does every polynomial factor? Why or why not?
6.
Objectives:
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Has it been factored completely?
Activities and Questions to ask students:
Factor trinomials of the form x + bx + c by unFOILing
2
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Factor out the GCF and then factor trinomials of the form
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Encourage students to look at this section as “brainteasers” to solve.
If we multiplied two binomials and received a trinomial, is there a way to get the two
binomials by factoring a trinomial? Look at x2 – x – 12. Think about how to factor. Have
students discuss in groups.
Elicit answers and ask how did you get that? Did you check to see if you factoring correctly?
How could you check?
Ask if someone else got it a different way. Emphasize the correct methods.
Do the same for another example like x2 – 27x + 50.
Show the backwards box:
x
a
x
(x)2
ax
b
bx
ab
Recall when multiplying binomials, ax + bx = the middle term. Ab = constant term
So, a + b = -27 and ab = 50. (x -25)(x-2)
What are other ways you can factor this? Can you recognize a pattern?
Have students work through another example: y2 – 3y – 40 and q2 +6q + 9. Talk about the
signs of these numbers and the significance in the answer.
Try another example: x2 + 6x + 15 – this is prime!
Whenever you ask students to factor, always ask is it factored completely?
Try another problem: 4x2 - 24x + 36 What is the difference with this problem? Make sure
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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students notice the 4 in front. What could you do with the 4?
Once you factor the 4, 4(x2 - 6x + 9), now what do you do? Is it factored completely?
Have students continue to factor so 4(x – 3)(x – 3) or 4(x – 3)2
Try another problem: x3 + 3x2 – 4x. Have students discuss how they would factor.
After factoring, ask is it factored completed? How do you know it’s factored completely?
Ask how they factored it? Elicit other ways from students. Discuss a pattern of how they
factored. First, factored out GCF, then they factored the trinomial.
Factoring Trinomials of the Form ax 2 + bx + c (2 days)
Overview of Objectives, students should be able to:
•
2
Factor trinomials of the form ax + bx + c , where
a ≠1
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Factor out the GCF before factoring a trinomial of the
form ax + bx + c
Main Overarching Questions:
1. What changes in how you factor when a ≠ 1
2. How do you remember to always check for a GCF before factoring any trinomial?
3. How can you check your final answer?
2
4.
Objectives:
•
Activities and Questions to ask students:
Factor trinomials of the form ax + bx + c , where
2
a ≠1
How do you know your polynomial is completely factored?
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With a problem like 4x2 - 24x + 36, the 4 was a GCF. What if you don’t have a GCF? How can
you factor it? For example: 3r2 +10r – 8 , answer: (3r – 2)(r + 4)
Ask students to think about how they would factor it? Discuss different factoring ideas.
Ask students to get into groups to talk about ways of factoring. How did they factor their
trinomial?
Have students discuss the logic behind their method of factoring.
Is it factored completely? Why or why not?
Is (3r-2)(r+4) = 3r – 2(r+4)? NO, Don’t make that mistake!!! Always put your parenthesis
correctly.
Try another one: 35x2 +4x – 4. Answer: (5x + 2)(7x – 2)
Try another one: 6x4 – 5x2 + 1. Answer: (3x2 – 1)(2x2 – 1). What was the difference with this
one? Is it factored completely?
Try one more: 12a2 – 16ab – 3b2. Answer: (6a + b)(2a – 3b)
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
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Factor out the GCF before factoring a trinomial of the
form ax + bx + c
2
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What are the types of strategies we used so far?
o Factoring out a GCF if there is a factor in common of all the terms.
o If a = 1, using the reverse box method (unFOIL)
o If a not equal to 1, (list methods students were using)
What would you do if you had a trinomial like this: 3x3 + 17x2 + 10x
What is the difference with that trinomial? (each term has an x)
Elicit methods students would use to factor. Ask students to factor and report their answers.
Talk through how they got their answer.
Did anyone else factor differently? Did you get the same answer? Why or why not?
Try another problem: 6xy2 + 33xy – 18x. This will take more time to answer 3x(2y-1)(y+6)
Try another problem: -5x2 – 19x + 4. Answer: -(x+4)(5x-1)
Ask students complete worksheets in groups. Report answers and have students come to a
consensus on answers.
Factoring Trinomials of the Form ax 2 + bx + c by Grouping (2 days)
Overview of Objectives, students should be able to:
1.
Use the grouping method to factor trinomials of the form
ax 2 + bx + c
Objectives:
•
Use the grouping method to factor trinomials of the form
ax + bx + c
Main Overarching Questions:
1. Why would you want to factor by grouping?
2.
When would you want to factor by grouping?
Activities and Questions to ask students:
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Recall before when we had four terms and we grouped terms together in order to
factor? Recall 16x3 – 28x2 - 12x – 21. Answer: (4x-7)(4x2+3)
Can we also use this method in factoring trinomials? How do you think we could use
this?
How could you separate the b so you can factor by grouping?
Try this example: 3x2 +14x + 8
Show them this method:
Factors of
Sums of the factor
(3 * 8) = 24
(sum to 14)
1, 24
25
2,12
14
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
3, 8
11
3x2 +14x + 8 = 3x2 + 12x + 2x + 8. How do you know which terms should be grouped
together?
Then, factor by grouping. 3x(x+4) + 2(x+4). Is it done?
Answer: (3x + 2)(x + 4)
• Can you think of another way to factor trinomials by grouping? What are they?
• Factor: 30x2 - 26x + 4.
• Factor: 6x2y – 7xy – 5y (Here they have to remember to factor out that y or else it is
not factored completely.
• Is it factored completely?
• Give students worksheets and discuss answers and methods students used to
completely factor the trinomials.
Factoring Perfect Square Trinomials and the Difference of Two Squares (1 day)
Overview of Objectives, students should be able to:
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Recognize perfect square trinomials
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Factor perfect square trinomials
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Factor the difference of two squares
Objectives:
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Recognize perfect square trinomials
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Factor perfect square trinomials
Main Overarching Questions:
1. How can you recognize a perfect square in order to factor?
2.
How can you recognize a difference of two squares?
Activities and Questions to ask students:
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Ask students to factor x2 + 12x +36, x2 + 20x + 100, and x2 + 18x + 81 and report their answers.
x2 + 12x +36 = (x+6)2, x2 + 20x + 100 = (x+10)2, and x2 + 18x + 81 = (x+9)2
What is the pattern? What is the relationship between the constant terms on each side of
the equations? 36 = 62; 100 = 102; 81 = 92;
What is the relationship between the middle term and the constant term on the other side?
12 = 2(6); 20 = 2(10); 18 = 2(9)
Recall when we multiplied binomials like: (a+b)2 = a2 +2ab +b2 or (x + 3)2 = x2 +6x+9
It would be greatly beneficial for students to be able to recognize Perfect Squares or
Difference of two Squares. Ask students to name the Perfect squares: 1, 4, 9, 16, 25, 36, 49,
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
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Factor the difference of two squares
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64, 81, 100, … Remembering these will be helpful.
Try factoring x2 – 10x + 25. Answer: (x – 5)2 Why do we SUBTRACT 5 here?
Try factoring 9r2 +24rs + 16s2. Can you use difference of squares here? Why or why not?
Answer: (3r + 4s)2
Factor 12x3 – 84x2 + 147x; answer: 3x(2x-7)2
Factor 4x2 + 9. What did you get? It’s prime. If students did give you an answer other than
itself, have them explain their answer and reasoning. Have students comment on why they
are incorrect constructively.
How would we factor 4x2 – 9?
Recall when we multiplied: (3x+y)(3x-y) We recognized the pattern as (a+b)(a-b) = a2 - b2
How could you use this information to factor 4x2 – 9? (2x-3)(2x+3)
Try factoring 9x5 – 25x3; What do you notice here that could help you factor? Answer: x3(3x5)(3x+5)
Try factoring -9x2 + 100. Have students talk about how they would factor. If recognize that
they have to factor out the negative first, Answer: -(3x-10)(3x+10) If they don’t remember
that, have them reorder the terms 100 – 9x2; answer (10 – 3x)(10+3x)
Is it factored completely?
Give students a worksheet to work in groups. This worksheet should include all the possible
factoring problems. Discuss answers. Have students give answers and defend their answers.
Always ask is it factored completely?
Solving Quadratic Equations by Factoring (2 days)
Overview of Objectives, students should be able to:
1.
Solve quadratic equations by factoring
2.
Solve equations with degree greater than 2 by factoring
Main Overarching Questions:
1. What does it mean to solve ax 2 + bx + c =
0
2. How many answers should you get?
3. How do you recognize that solving this equation, x-intercepts, and the zeros are the same?
4. Why do you factor and set each factor equal to zero?
5. Does you answer make sense?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
6.
Objectives:
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Solve quadratic equations by factoring
Activities and Questions to ask students:
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Solve equations with degree greater than 2 by factoring
Here we expect SOLUTIONS since there is an equal sign.
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Ask students to suppose they are told that the product of two factors is 12. Do they know for
certain what either factor is? Why or why not?
Now, the product of two factors is 0. Do you know for certain what either factor is?
Ask if you have an equation like ab = 0, what can a and b equal?
Have students talk about the implications on whether a = 0 and/or b = 0.
How could you solve an equation like (x + 2) (x-3) = 0? {-2, 3}
Why would you set each factor equal to zero?
What would be the solution to (x-10)(3x+1) = 0? {10, -1/3}
What would you do if you have an equal like x2-4x+3 = 0? How could you solve this equation?
{1,3}
How could you check your answer?
What is the solution to x(3x+7)=6? How would you solve this equation? What is the first
thing you should do? Why? {2/3 , -3}
How many answers should you get? How many answers do you expect to get when you have
a linear equation? Does the degree of the equation have anything to do with the expected
number of solutions? Why or why not? What do you think the connection is?
Try solving 2x3 – 18x = 0 . How many solutions should you get? (3) {0, 3, -3}
Whenever you have an equations with a degree higher than 1, you should factor. Always
factor.
Solve (x + 3)(3x2 – 20x – 7) = 0; { -3, -1/3, 7} Have students talk about how they solve this.
How would you solve this equation? Why?
Quadratic Equations and Problem Solving (2 days)
Overview of Objectives, students should be able to:
1.
Solve problems that can be modeled by quadratic
equations.
Main Overarching Questions:
1. When will the ball hit the ground?
2. What type of problems can be represented using a quadratic equation?
3. Does your answer make sense? Can you justify it as being correct?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
Objectives:
•
Solve problems that can be modeled by quadratic
equations.
Activities and Questions to ask students:
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Look a ball projectile problem. Since students haven’t graphed yet, graph it to show them
the projectile of the ball.
Have students discuss the meaning behind the graph. When would the height of the ball be
at its peak?
When would the ball hit the ground?
Where the ball hits the ground, what do we call that on the graph?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.