WELCOME!!! Warm Up
• Turn in your polynomial adding and subtracting
worksheet
• Take a piece of paper and fold it into thirds to
make a name card. Put your name on the card.
• Divide using long division. No calculator!
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32 )435028358602
Video – Not the best, but why not?
Synthetic Division: http://www.youtube.com/watch?v=Ar10yTC9W8M
Objectives
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Multiply polynomials using distributive property
Multiply two binomials using the FOIL method
Multiply polynomials using the box method
Use the Binomial Theorem (Pascals Triangle) to
expand binomials raised to powers
• Divide polynomials using long division and
synthetic division
• Apply these processes to abstract problems
Today:
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•
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Multiplication Lesson
Multiplication Worksheet
Pascals Triangle Worksheet
Long Division and Synthetic Division Lesson
Division Worksheet
Game
Review
Post Assessment
Do you agree…?
3(4+4) = 3(8)
= 24
3(4+4) = 3(4) + 3(4)
= 12 + 12
= 24
Why does this happen?
Vocab: D_ _ _ _ _ _ _ _ _
Remember this… we will
use it very soon!
Multiplying Monomials
Multiply coefficients with coefficients (number
next to the variables) and powers of x with
powers of x [remember we add exponents
when we multiply numbers with same base….
Why? X3X4 = (x x x) (x x x x) = (x x x x x x x) = x7 ]
Try: (4x3)(2x)
(remember x is the same as x1)
Try: (15x100)(10x300)
Independent Practice – in your notes
1. (4x)(5x)=
2. (-3x)(8x2)=
3. (12x)(2x3)=
4.
(8x)(2x2)(2x4)=
Independent Practice – Answers
1. (4x)(5x)= 20x2
2. (-3x)(8x2)= -24x3
3. (12x)(2x3)= 24x4
4. (8x)(2x2)(2x4)= 32x7
Distributive Property and Polynomials
• Distributive Property: a(b + c) = ab + ac
Independent Practice
5. 3(x2 + 2x + 9) =
6. 4x(x3 + x - 100) =
7. -10x3(2x + 8) =
8. 20x3(4x2 + 8x + 6) =
Independent Practice - Answers
5. 3(x2 + 2x + 9) = 3x2 + 6x + 27
6. 4x(x3 + x - 100) = 4x4 + 4x2 - 400x
7. -10x3(2x + 8) = -20x4 - 80x3
8. 20x3(4x2 + 8x + 6) = 80x5 + 160x4 + 120x3
Multiplying Polynomials
Step 1) Distribute monomials of first polynomial
through each term of the second polynomial
Step 2) Multiply the new terms out
Step 3) Combine like terms
Step 4) Order the terms into Standard Form
FOIL Method
• When multiplying 2 binomials you can FOIL to
remember how to distribute
F
O
I
L
Firsts / Fronts
Outside
Inside
Last
(y + 3)(y + 7).
F tells you to multiply the FIRST terms
of each binomial.
y2
(y + 3)(y + 7).
O tells you to multiply the OUTER
terms of each binomial.
y2 + 7y
(y + 3)(y + 7).
I tells you to multiply the INNER
terms of each binomial.
y2 + 7y + 3y
(y + 3)(y + 7).
L tells you to multiply the LAST terms
of each binomial.
y2 + 7y + 3y + 21
Combine like terms.
y2 + 10y + 21
Box Method
Independent Practice
9. (4x+2)(2x+3) =
10. (5x2 – 3)(2x+8)
11. (2x+5) (2x – 5)
12. (3x+5)(3x+5)
Independent Practice - Answers
9. (4x+2)(2x+3) = 8x2 + 12x + 4x + 6 = 8x2 + 16x +6
10. (5x2 – 3)(2x+8) = 10x3 +40x2 - 6x -24
11. (2x+5) (2x – 5) = 4x2 - 25
12.(3x+5)(3x+5) = 9x2 + 15x + 15x + 25
= 9x2 +30x +25
Multiplying Worksheet
Pascals Triangle
?
?
?
?
Binomial Theorem
• You can expand any (x+y)n using the binomial
theorem
(x+y)2 = x2 + 2xy + y2 (how many terms do we have?)
(x+y)3 = x3 + 3x2y + 3xy2 + y3 (how many terms do we have?)
Do you see any patterns?
How does this relate to pascal’s triangle?
Begin working on Exploratory Worksheet
Special Patterns?
(x + y)2 = x2 + 2xy + y2
(x - y)2 = x2 - 2xy + y2
(x + y)(x - y) = x2 - y2
(x + a)(x + b) = x2 + (a + b)x + ab
(x + y)3 = x3 + 3 x2y + 3xy2 + y3
(x - y)3 = x3 - 3x2y + 3xy2 – y3
Long Division
• Do we remember the steps from the warm up
earlier?
• Can we apply them using polynomial terms?
Long Division Polynomial Problem
Set it up:
How did we do?
Independent Practice
13. (m3 – 20) / (m – 3)
14. (5x3 + 11x2 + 26x + 26) / (5x + 6)
Independent Practice - Solutions
13. (m3 – 20) / (m – 3)
m2 +3m
7
+9+
M-3
14.(5x3 + 11x2 + 26x + 26) / (5x + 6)
x2
2
+x+4+
5x+6
Synthetic Division
• Synthetic Division is an algorithm for when we
already know a factor
• We can use this method to divide by binomials of
the form (x-a)
• Its important to note that the factor must be
(x-a)
• Examples: (x-2), (x+5), (x - 100)
NOT (2x-4) , (4x-5), or (20x + 3)
Long Division Problem
Set it up:
The number
that would
solve the
divisor to zero
All the coefficients of the
polynomial dividend terms from
highest to lowest (standard form)
NOTE: make sure to
include any terms
that may have been
omitted: 0x2 or 0x
wouldn’t show up in
a polynomial of
standard form, but
we need it as a
placeholder
Long Division Problem Work
How did we do?
Division Worksheet
Game Preparation
• Take 1 blank piece of
paper: Fold your
paper once hot-dog
and in thirds
hamburger creating 6
boxes on your page.
• Number the boxes 1-6
1
2
3
4
5
6
Game Rules/ Procedures
• We will do 6 polynomial review questions
• This is an independent game, everyone participates.
You will turn final page in for grade.
• Once a questions is flashed, we will begin the timers.
• Points will be awarded for the right answer in the
quickest time.
• A wrong answer earns you no points, but you are still
required to complete the problem for the final
submission after the game!
• (Do your best at your own pace for best results)
Sample Question Format
1.
2.
3.
4.
What class is this?
What units have we covered?
Y/N Ms. Justin’s Videos were awesome!?
Am I ready to play?
:30
5pts
1:00
4pts
1:30
3pts
2:00
1pt
Question 1:
Give an example of….
1.
2.
3.
4.
:30
5pts
cubic, trinomial
quadratic binomial
constant, monomial
polynomial
1:00
4pts
1:30
3pts
2:00
1pt
Question 2:
1. In words, explain how to subtract polynomial
2. (3x3 + 4x2 – 5x + 8) – (2x3 – 8x2 + 4x + 5)
:30
5pts
1:00
4pts
1:30
3pts
2:00
1pt
Question 3:
1. What does distribute mean?
2. Write as a polynomial in standard form:
5x (4x2 – 5x + 3x3+ 8)
:30
5pts
1:00
4pts
1:30
3pts
2:00
1pt
Question 4:
1. What does F O I L stand for?
2. What kind of polynomials can we FOIL?
3. Find the area of the rectangle:
(3x + 8)
(4x2- 5)
:30
5pts
1:00
4pts
1:30
3pts
2:00
1pt
Question 5:
• What is this pattern?
• What comes next?
• How do we use this
when multiplying
polynomials?
:30
5pts
1:00
4pts
1:30
3pts
1
1 1
1 2 1
1 3 3 1
2:00
1pt
Question 6
Divide
1:00
10pts
2:00
8pts
4:00
5pt
6:00
2pt
GAME OVER!
WHO WON?
HAVE WE MET THE OBJECTIVES?
ARE WE READY FOR THAT ASSESSMENT?
LAST QUESTIONS?
5 MINUTES TO STUDY:
• Polynomial vocabulary
• Adding and Subtracting
• Multiplying
– Distributing, FOIL, special patterns, Pascal
• Dividing
– Long, synthetic
POST ASSESSMENT
GOODLUCK