Unit 6

1
Unit 6 Chapter 9
2
Unit 6

• Chapter 8 Review and Chap. 8 Skills
• Section 9.1 Adding and Subtracting Polynomials
• Section 9.2 Multiply Polynomials
• Section 9.3 Special Products of Polynomials
• Section 9.4 Solve Polynomial Equations
• Section 9.5 Factor x2 bx c
• Section 9.6 - Factor ax2 bx c
• Section 9.7 and 9.8 Factoring Special Products
  and Factoring Polynomials Completely

3
Warm-Up X.X
4
Vocabulary X.X

• Holder
• Holder 2
• Holder 3
• Holder 4

5
Notes X.X LESSON TITLE.

• Holder
• Holder
• Holder
• Holder
• Holder

6
Examples X.X

7
Warm-Up Chapter 9
8
Prerequisite Skills
SKILL CHECK
Simplify the expression.
7. 3x ( 6x)
8. 5 4x 2
9. 4(2x 1) x
10. (x 4) 6 x
9
Prerequisite Skills
SKILL CHECK
Simplify the expression.
11. (3xy)3
13. (x5)3
14. ( x)3
10
Vocabulary 9.1

• Degree of a polynomial
• Term with highest degree
• Leading Coefficient
• Coefficient of the highest degree term
• Binomial
• Polynomial with 2 terms
• Trinomial
• Polynomial with 3 terms

• Monomial
• Number, variable, or product of them
• Degree of a Monomial
• Sum of the exponents of the variables in a term
• Polynomial
• Monomial or Sum of monomials with multiple terms

11
Notes 9.1 - Polynomials

• CLASSIFYING POLYNOMIALS
• What is NOT a polynomial? Terms with
• Negative exponents
• Fractional exponents
• Variables as exponents
• EVERYTHING ELSE IS A POLYNOMIAL!
• To find DEGREE of a term
• Add the exponents of each variable
• Mathlish Polynomial Grammar
• All polynomials are written so that the degree of
  the exponents decreases (i.e. biggest first)

12
Notes 9.1 Polynomials Cont.

• I can only combine things in math that ????
• ADDING POLYNOMIALS
• Combine like terms
• Remember to include the signs of the
  coefficients!
• SUBTRACTING POLYNOMIALS
• Use distributive property first!!
• Combine like terms
• Remember to include the signs of the
  coefficients!

13
Examples 9.1

14

EXAMPLE 1
Rewrite a polynomial
Write 15x x3 3 so that the exponents decrease
from left to right. Identify the degree and
leading coefficient of the polynomial.
SOLUTION
Consider the degree of each of the polynomials
terms.
15x x3 3
The polynomial can be written as x3 15x 3.
The greatest degree is 3, so the degree of the
polynomial is 3, and the leading coefficient is
1.
15
EXAMPLE 2
Identify and classify polynomials
Tell whether is a polynomial. If it is a
polynomial, find its degree and classify it by
the number of its terms. Otherwise, tell why it
is not a polynomial.
16
EXAMPLE 3
Add polynomials
Find the sum.
a. (2x3 5x2 x) (2x2 x3 1)
b. (3x2 x 6) (x2 4x 10)
17
EXAMPLE 3
Add polynomials
SOLUTION
a. Vertical format Align like terms in
vertical columns.
(2x3 5x2 x)
3x3 3x2 x 1
b. Horizontal format Group like terms and
simplify.
(3x2 x 6) (x2 4x 10)
(3x2 x2) (x 4x) ( 6 10)
4x2 5x 4
18

EXAMPLE 1
Rewrite a polynomial
for Examples 1,2, and 3
GUIDED PRACTICE
SOLUTION
Consider the degree of each of the polynomials
terms.
5y 2y2 9
The polynomial can be written as 2y2 5y 9.
The greatest degree is 2, so the degree of the
polynomial is 2, and the leading coefficient is 2
19
EXAMPLE 2
for Example
for Examples 1,2, and 3
Identify and classify polynomials
GUIDED PRACTICE
SOLUTION
y3 4y 3 is a polynomial. 3 degree trinomial.
20
EXAMPLE 3
for Example
for Examples 1,2, and 3
Add polynomials
GUIDED PRACTICE
a. (2x3 4x x) (4x2 3x3 6)
21
EXAMPLE 3
for Example
for Examples 1,2, and 3
Add polynomials
GUIDED PRACTICE
SOLUTION
a. Vertical format Align like terms in
vertical columns.
(5x3 4x 2x)
8x3 4x2 2x 6
b. Horizontal format Group like terms and
simplify.
(5x3 4x 2) (4x2 3x3 6)
8x3 4x2 2x 6
22
EXAMPLE 4
Subtract polynomials
Find the difference.
a. (4n2 5) ( 2n2 2n 4)
b. (4x2 3x 5) (3x2 x 8)
23
EXAMPLE 4
Subtract polynomials
SOLUTION
a. (4n2 5)
4n2 5
6n2 2n 9
b. (4x2 3x 5) (3x2 x 8)
4x2 3x 5 3x2 x 8
(4x2 3x2) ( 3x x) (5 8)
x2 2x 13
24
EXAMPLE 4
for Examples 4 and 5
Subtract polynomials
GUIDED PRACTICE
a. (4x2 7x) ( 5x2 4x 9)
(4x2 7x ) (5x2 4x 9)
4x2 7x 5x2 4x 9
(4x2 5x2) ( 7x 4x) 9
x2 11x 9
25
EXAMPLE 5
Solve a multi-step problem
M ( 488t2 5430t 24,700) ( 318t2 3040t
25,600)
( 488t2 318t2) (5430t 3040t) (24,700
25,600)
170t2 2390t 900
Substitute 6 for t in the model, because 2001 is
6 years after 1995.
26
EXAMPLE 5
Solve a multi-step problem
27
Warm-Up 9.2
28
Lesson 9.2, For use with pages 561-568
1. Simplify 2 (9a b).
4. Simplify x2(x1) 2x(3x3) 2x 5
3. Simplify 2x(3x 2)
29
Lesson 9.2, For use with pages 561-568
3. The number of hardback h and paperback p books
(in hundreds) sold from 19992005 can be modeled
by h 0.2t2 1.7t 14 and p 0.17t3 2.7t2
11.7t 27 where t is the number of years since
1999. About how many books sold in 2003.
4. Simplify (x 1)(x 2)
30
Vocabulary 9.2

• Polynomial
• Monomial or Sum of monomials with multiple terms

31
Notes 9.2 Multiply Polynomials

• Multiplying Polynomials is like using the
  distributive property over and over and over
  again.
• Everything must be multiplied by everything else
  and combine like terms!!!
• Frequently people use the FOIL process to
  multiply polynomials.
• F Multiply the First Terms
• O Multiply the Outside Terms
• I Multiply the Inside Terms
• L Multiply the Last Terms

32
Examples 9.2

33

EXAMPLE 1
Multiply a monomial and a polynomial
Find the product 2x3(x3 3x2 2x 5).
2x3(x3 3x2 2x 5)
Write product.
2x3(x3) 2x3(3x2) 2x3(2x) 2x3(5)
Distributive property
2x6 6x5 4x4 10x3
Product of powers property
34
EXAMPLE 2
Multiply polynomials using a table
Find the product (x 4)(3x 2).
SOLUTION
STEP 1
Write subtraction as addition in each polynomial.
(x 4)(3x 2) x ( 4)(3x 2)
35
EXAMPLE 2
Multiply polynomials using a table
STEP 2
Make a table of products.
36
for Examples 1 and 2
GUIDED PRACTICE
Find the product.
SOLUTION
x(7x2 4)
Write product.
x(7x2 )x(4)
Distributive property
7x34x
Product of powers property
37
for Examples 1 and 2
GUIDED PRACTICE
Find the product.
SOLUTION
Make a table of products.
38
for Examples 1 and 2
GUIDED PRACTICE
Find the product.
SOLUTION
STEP 1
Write subtraction as addition in each polynomial.
(4n 1) (n 5) 4n ( 1)(n 5)
39
for Examples 1 and 2
GUIDED PRACTICE
STEP 2
Make a table of products.
40

EXAMPLE 3
Multiply polynomials vertically
Find the product (b2 6b 7)(3b 4).
SOLUTION
3b3 14b2 45b 28
41
EXAMPLE 4
Multiply polynomials horizontally
Find the product (2x2 5x 1)(4x 3).
Solution Multiply everything and get (2x2)(4x)
(2x2)(-3) (5x)(4x) (5x)(-3) (-1)(4x)
(-1)(-3) 8x3 14x2 19x 3
FOIL PATTERN The letters of the word FOIL can
help you to remember how to use the distributive
property to multiply binomials. The letters
should remind you of the words First, Outer,
Inner, and Last.
(2x 3)(4x 1)
8x2 2x 12x 3
42
EXAMPLE 5
Multiply binomials using the FOIL pattern

Find the product (3a 4)(a 2).
(3a 4)(a 2)
(3a)(a) (3a)( 2) (4)(a) (4)( 2)
Write products of terms.
3a2 ( 6a) 4a ( 8)
Multiply.
3a2 2a 8
Combine like terms.
43
for Examples 3, 4, and 5
GUIDED PRACTICE
Find the product.
SOLUTION
x3 4x2 5x 2
44
for Examples 3, 4, and 5
GUIDED PRACTICE
Find the product.
SOLUTION
(3y2 y 5)(2y 3)
Write product.
3y2(2y 3) y(2y 3) 5(2y 3)
Distributive property
6y3 9y2 2y2 3y 10y 15
Distributive property
6y3 11y2 13y 15
Combine like terms.
45
for Examples 3, 4, and 5
GUIDED PRACTICE
Find the product.
SOLUTION
(4b)(b) (4b)( 2) (5)(b) (5)( 2)
Write products of terms.
4b2 8b 5b 10
Multiply.
4b2 13b 10
Combine like terms.
46
EXAMPLE 6
Standardized Test Practice
The dimensions of a rectangle are x 3 and x
2. Which expression represents the area of the
rectangle?
SOLUTION
Formula for area of a rectangle
(x 3)(x 2)
Substitute for length and width.
x2 2x 3x 6
Multiply binomials.
47
EXAMPLE 6
Standardized Test Practice
x2 5x 6
Combine like terms.
48
Warm-Up 9.3
49
Lesson 9.3, For use with pages 569-574
Find the product.
1. (x 7)(x 7)
3. (x 7)(x - 7)
2. (x - 7)(x - 7)
2. (3x 1)(3x 2)
50
Lesson 9.3, For use with pages 569-574
Find the product.
3. The dimensions of a rectangular playground can
be represented by 3x 8 and 5x 2. Write a
polynomial that represents the area of the
playground. What is the area of the playground
if x is 8 meters?
51
Vocabulary 9.3

• Binomial
• Polynomial with 2 terms
• Trinomial
• Polynomial with 3 terms

52
Notes 9.3 Special Products of Poly.

• Find the area of the larger square
• Multiply (ab)2
• (ab)(ab)
• a2 2ab b2
• Multiply (a b) 2
• (a b)(a b) a2 - 2ab b2

53
Notes 9.3 Special Products of Poly.

• Multiply (a b)(a - b)
• a2 - b2
• This is a very special type of polynomial called
  the DIFFERENCE OF TWO SQUARES

54
Examples 9.4

55
Use the square of a binomial pattern
EXAMPLE 1
Find the product.
a. (3x 4)2
(3x)2 2(3x)(4) 42
Square of a binomial pattern
9x2 24x 16
Simplify.
b. (5x 2y)2
(5x)2 2(5x)(2y) (2y)2
Square of a binomial pattern
Simplify.
25x2 20xy 4y2
56
for Example 1
GUIDED PRACTICE
Find the product.
1. (x 3)2
(x)2 2(x)(3) (3)2
Square of a binomial pattern
Simplify.
x2 6x 9
2. (2x 1)2
Square of a binomial pattern
(2x)2 2(2x)(1) (1)2
Simplify.
4x2 4x 1
57
for Example 1
GUIDED PRACTICE
3. (4x y)2
(4x)2 2(4x)(y) (y)2
Square of a binomial pattern
Simplify.
16x2 8xy y2
4. (3m n)2
(3m)2 2(3m)(n) (n)2
Square of a binomial pattern
Simplify.
9m2 6mn n2
58

EXAMPLE 2
Use the sum and difference pattern
Find the product.
a. (t 5)(t 5)
t2 52
Sum and difference pattern
t2 25
Simplify.
b. (3x y)(3x y)
(3x)2 y2
Sum and difference pattern
9x2 y2
Simplify.
59
for Example 2
GUIDED PRACTICE
Find the product.
5. (x 10)(x 10)
x2 102
Sum and difference pattern
x2 100
Simplify.
6. (2x 1)(2x 1)
(2x)2 12
Sum and difference pattern
4x2 1
Simplify.
7. (x 3y)(x 3y)
(x)2 (3y)2
Sum and difference pattern
x2 9y2
Simplify.
60
EXAMPLE 3
Use special products and mental math
SOLUTION
Notice that 26 is 4 less than 30 while 34 is 4
more than 30.
(30 4)(30 4)
Write as product of difference and sum.
302 42
Sum and difference pattern
900 16
Evaluate powers.
884
Simplify.
61
EXAMPLE 4
Solve a multi-step problem
Border Collies
The color of the dark patches of a border
collies coat is determined by a combination of
two genes. An offspring inherits one patch color
gene from each parent. Each parent has two color
genes, and the offspring has an equal chance of
inheriting either one.
62
EXAMPLE 4
Solve a multi-step problem
The gene B is for black patches, and the gene r
is for red patches. Any gene combination with a B
results in black patches. Suppose each parent has
the same gene combination Br. The Punnett square
shows the possible gene combinations of the
offspring and the resulting patch color.
63
EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Notice that the Punnett square shows 4 possible
gene combinations of the offspring. Of these
combinations, 3 result in black patches.
64
EXAMPLE 4
Solve a multi-step problem
STEP 2
Model the gene from each parent with 0.5B
0.5r. There is an equal chance that the collie
inherits a black or red gene from each parent.
The possible genes of the offspring can be
modeled by (0.5B 0.5r)2. Notice that this
product also represents the area of the Punnett
square.
Expand the product to find the possible patch
colors of the offspring.
(0.5B 0.5r)2 (0.5B)2 2(0.5B)(0.5r) (0.5r)2
0.25B2 0.5Br 0.25r2
65

EXAMPLE 4
Solve a multi-step problem
Consider the coefficients in the polynomial.
0.25B2 0.5Br 0.25r2
The coefficients show that 25 50 75 of the
possible gene combinations will result in black
patches.
66
for Examples 3 and 4
GUIDED PRACTICE
8. Describe how you can use special product to
find 212.
Use the square of binomial pattern to find the
product (20 1)2.
(20 1)2 (20)2 2(20)(1) 12
Square of a binomial pattern
421
Simplify.
67
for Examples 3 and 4
GUIDED PRACTICE
BORDER COLLIES
Look back at Example 4. What percent of the
possible gene combinations of the offspring
result in red patches?
SOLUTION
STEP 1
Notice that the Punnett square shows 4 possible
gene combinations of the offspring. Of these
combinations, 1 result in red patches.
68
for Examples 3 and 4
GUIDED PRACTICE
STEP 2
Model the gene from each parent with 0.5B
0.5r. There is an equal chance that the collie
inherits a black or red gene from each parent.
The possible genes of the offspring can be
modeled by (0.5B 0.5r)2. Notice that this
product also represents the area of the Punnett
square.
Expand the product to find the possible patch
colors of the offspring.
(0.5B 0.5r)2 (0.5B)2 2(0.5B)(0.5r) (0.5r)2
0.25B2 0.5Br 0.25r2
69
for Examples 3 and 4
GUIDED PRACTICE
Consider the coefficients in the polynomial.
0.25B2 0.5Br 0.25r2
The coefficients show that 25 of the possible
gene combinations will result in red patches.
70
Warm-Up 9.4
71
Lesson 9.4, For use with pages 575-580
1. Find the GCF of 12 and 28.

• Find the binomials that
• multiply to get x2 - 9

2. (2a 5b)(2a 5b)
72
Lesson 9.4, For use with pages 575-580

• Write down an equation with values for X and Y
  that make the following equation true
• X Y 0

73
Vocabulary 9.4

• Roots
• Solutions for x when a function 0
• Also where the function crosses the X axis

74
Notes 9.4 Solve Polynomial Eqns.

• SOLVING EQUATIONS THAT EQUAL ZERO
• Use the Zero Product Property to solve algebraic
  equations that equal 0.
• Solutions to algebraic equations that equal zero
  are called the roots or the zeros of a
  function.
• FACTORING
• To solve a polynomial equation, it may be
  necessary to break it up into its factors.
• Find the GCF of ALL the terms and factor it out.
  Its like unmultiplying the distributive
  property.

75
Notes 9.4 Solve Polynomial Eqns.

• VERTICAL MOTION MODEL
• The height of an object (in FEET!!) can be
  modeled by the following equation
• H(t) -16t2 vt s

76
Examples 9.4

77

EXAMPLE 1
Use the zero-product property
Solve (x 4)(x 2) 0.
(x 4)(x 2) 0
Write original equation.
x 4 0
or
x 2 0
Zero-product property
x 4
or
x 2
Solve for x.
78
for Example 1
GUIDED PRACTICE
1. Solve the equation (x 5)(x 1) 0.
(x 5)(x 1) 0
Write original equation.
x 5 0
or
x 1 0
Zero-product property
x 5
or
x 1
Solve for x.
79
EXAMPLE 2
Find the greatest common monomial factor
Factor out the greatest common monomial factor.
a. 12x 42y
SOLUTION
80
EXAMPLE 2
Find the greatest common monomial factor
Factor out the greatest common monomial factor.
b. 4x4 24x3
SOLUTION
81
for Example 2
GUIDED PRACTICE
2. Factor out the greatest common monomial
factor from 14m 35n.
SOLUTION
The GCF of 14 and 35 is 7. The variables m and n
have no common factor. So, the greatest common
monomial factor of the terms is 7.
82
EXAMPLE 3
Solve an equation by factoring
Solve 2x2 8x 0.
2x2 8x 0.
Write original equation.
2x(x 4) 0
Factor left side.
or
x 4 0
2x 0
Zero-product property
or
x 0
x 4
Solve for x.
83
EXAMPLE 4
Solve an equation by factoring
Solve 6n2 15n.
6n2 15n 0
Subtract 15n from each side.
3n(2n 5) 0
Factor left side.
or
2n 5 0
3n 0
Zero-product property
or
n 0
Solve for n.
84
for Examples 3 and 4
GUIDED PRACTICE
Solve the equation.
3. a2 5a 0.
a2 5a 0
Write original equation.
a(a 5) 0
Factor left side.
or
a 5 0
a 0
Zero-product property
or
a 0
a 5
Solve for x.
85
for Examples 3 and 4
GUIDED PRACTICE
4. 3s2 9s 0.
3s2 9s 0
Write original equation.
3s(s 3) 0
Factor left side.
or
s 3 0
3s 0
Zero-product property
or
s 0
s 3
Solve for x.
86
for Examples 3 and 4
GUIDED PRACTICE
5. 4x2 2x.
4x2 2x
Write original equation.
4x2 2x 0
Subtract 2x from each side.
2x(2x 1) 0
Factor left side.
or
2x 1 0
2x 0
Zero-product property
or
x 0
Solve for x.
87
EXAMPLE 5
Solve a multi-step problem
ARMADILLO
A startled armadillo jumps straight into the air
with an initial vertical velocity of 14 feet per
second.After how many seconds does it land on the
ground?
88
EXAMPLE 5
Solve a multi-step problem
SOLUTION
STEP 1
Write a model for the armadillos height above
the ground.
h 16t2 vt s
Vertical motion model
h 16t2 14t 0
Substitute 14 for v and 0 for s.
h 16t2 14t
Simplify.
89
EXAMPLE 5
Solve a multi-step problem
STEP 2
Substitute 0 for h. When the armadillo lands, its
height above the ground is 0 feet. Solve for t.
0 16t2 14t
Substitute 0 for h.
0 2t(8t 7)
Factor right side.
2t 0
8t 7 0
or
Zero-product property
or
t 0
t 0.875
Solve for t.
90
for Example 5
GUIDED PRACTICE
6. WHAT IF? In Example 5, suppose the initial
vertical velocity is 12 feet per second.After
how many seconds does armadillo land on the
ground?
SOLUTION
STEP 1
Write a model for the armadillos height above
the ground.
h 16t2 vt s
Vertical motion model
h 16t2 12t 0
Substitute 12 for v and 0 for s.
h 16t2 12t
Simplify.
91
for Example 5
GUIDED PRACTICE
STEP 2
Substitute 0 for h. When the armadillo lands, its
height above the ground is 0 feet. Solve for t.
0 16t2 12t
Substitute 0 for h.
0 4t(4t 3)
Factor right side.
4t 0
4t 3 0
or
Zero-product property
or
t 0
t 0.75
Solve for t.
92
Warm-Up 9.5
93
Lesson 9.5, For use with pages 582-589
Find the product.
1. (y 3)(y - 5)
3. (y - 3)( y - 5)
2. (y 3)( y 5)
4. (2y 3)( y 5)
94
Vocabulary 9.5

• Zero of a Function
• The X value(s) where a function equals zero
• AKA the roots of a function
• Factoring a Polynomial
• Finding the polynomial factors that will multiply
  to get the original.
• Its unFOILing or unmultiplying a polynomial

95
Notes 9.5 Factor x2bxc

• If you multiply two binomials (x p)(x q) to
  get x2 bx c, the following must be true
• p q c
• p q b (NOTICE ALL X COEFF. ARE 1!!)
• We can use these truths to go the other
  direction, and factor a polynomial into binomials

(x p)(x q) Polynomial x2 bx c Signs of b and c
(x 2)(x 3) x2 5x 6 b and c positive
(x 2)(x - 3) x2 - x 6 b and c negative
(x - 2)(x 3) x2 x 6 b is pos., c is neg.
(x - 2)(x - 3) x2 - 5x 6 B is neg., c is pos.
96
Examples 9.5

97
EXAMPLE 1
Factor when b and c are positive
Factor x2 11x 18.
SOLUTION
Find two positive factors of 18 whose sum is 11.
Make an organized list.
98

EXAMPLE 1
Factor when b and c are positive
The factors 9 and 2 have a sum of 11, so they are
the correct values of p and q.
CHECK
(x 9)(x 2)
x2 2x 9x 18
Multiply binomials.
Simplify.
99
for Example 1
GUIDED PRACTICE
Factor the trinomial
1. x2 3x 2.
SOLUTION
Find two positive factors of 2 whose sum is 3.
Make an organized list.
100
EXAMPLE 1
Factor when b and c are positive
for Example 1
GUIDED PRACTICE
The factors 2 and 1 have a sum of 3, so they are
the correct values of p and q.
101
for Example 1
GUIDED PRACTICE
Factor the trinomial
2. a2 7a 10.
SOLUTION
Find two positive factors of 10 whose sum is 7.
Make an organized list.
102
EXAMPLE 1
Factor when b and c are positive
for Example 1
GUIDED PRACTICE
The factors 5 and 2 have a sum of 7, so they are
the correct values of p and q.
103
for Example 1
GUIDED PRACTICE
Factor the trinomial
3. t2 9t 14.
SOLUTION
Find two positive factors of 14 whose sum is 9.
Make an organized list.
104
EXAMPLE 1
Factor when b and c are positive
for Example 1
GUIDED PRACTICE
The factors 7 and 2 have a sum of 9, so they are
the correct values of p and q.
105
EXAMPLE 2
Factor when b is negative and c is positive
Factor n2 6n 8.
Because b is negative and c is positive, p and q
must both be negative.
106
EXAMPLE 3
Factor when b is positive and c is negative
Factor y2 2y 15.
Because c is negative, p and q must have
different signs.
107
for Examples 2 and 3
GUIDED PRACTICE
Factor the trinomial
4. x2 4x 3.
Because b is negative and c is positive, p and q
must both be negative.
108
for Examples 2 and 3
GUIDED PRACTICE
Factor the trinomial
5. t2 8t 12.
Because b is negative and c is positive, p and q
must both be negative.
109
for Examples 2 and 3
GUIDED PRACTICE
Factor the trinomial
6. m2 m 20.
Because c is negative, p and q must have
different signs.
110
for Examples 2 and 3
GUIDED PRACTICE
Factor the trinomial
7. w2 6w 16.
Because c is negative, p and q must have
different signs.
111
EXAMPLE 4
Solve a polynomial equation
Solve the equation x2 3x 18.
x2 3x 18
Write original equation.
x2 3x 18 0
Subtract 18 from each side.
(x 6)(x 3) 0
Factor left side.
or
x 6 0
x 3 0
Zero-product property
or
x 3
x 6
Solve for x.
112
EXAMPLE 4
for Example 4
Solve a polynomial equation
GUIDED PRACTICE
8. Solve the equation s2 2s 24.
s2 2s 24.
Write original equation.
s2 2s 24 0
Subtract 24 from each side.
(s 4)(s 6) 0
Factor left side.
or
s 4 0
s 6 0
Zero product property
or
s 6
s 4
Solve for x.
113
EXAMPLE 5
Solve a multi-step problem
Banner Dimensions
You are making banners to hang during school
spirit week. Each banner requires 16.5 square
feet of felt and will be cut as shown. Find the
width of one banner.
SOLUTION
STEP 1
Draw a diagram of two banners together.
114
EXAMPLE 5
Solve a polynomial equation
STEP 2
Write an equation using the fact that the area of
2 banners is 2(16.5) 33 square feet. Solve the
equation for w.
Formula for area of a rectangle
Substitute 33 for A and (4 w 4) for l.
0 w2 8w 33
Simplify and subtract 33 from each side.
0 (w 11)(w 3)
Factor right side.
or
w 3 0
w 11 0
Zero-product property
w 11
w 3
or
Solve for w.
115
for Example 5
GUIDED PRACTICE
Write an equation using the fact that the area of
2 banners is 2(10) 20 square feet. Solve the
equation for w.
Formula for area of a rectangle
Substitute 20 for A and (4 w 4) for l.
20 w2 8w
Simplify
0 w2 8w 20
Subtract 20 from each side.
0 (w 10)(w 2)
Factor right side.
w 10 0
or
w 2 0
Zero-product property
w 10
or
w 2
Solve for w.
116
Warm-Up 9.6
117
Lesson 9.6, For use with pages 592-599
Factor and solve the polynomials.
Find the product.
1. (3c 3)(2c 3)
3. x2 x 6 0
2. (2y 3)(2y 1)
4. x2 13x -36
118
Lesson 9.6, For use with pages 592-599
Find the product.
3. A cat leaps into the air with an initial
velocity of 12 feet per second to catch a speck
of dust, and then falls back to the floor. How
long does the cat remain in the air? H(t)
-16t2 vt s
119
Vocabulary 9.6

• Trinomial
• Polynomial with 3 terms

120
Notes 9.6 Factor ax2bxc

• If you multiply two binomials (dx p)(ex q)
  to get ax2 bx c, the following must be true
• d e a (NOTICE X COEFF. DO NOT 1!!)
• p q c
• TO FACTOR POLYNOMIALS WHERE A ? 1
• If A -1, factor out -1 from the polynomial and
  factor
• If A gt 0, use a table to organize your work

Factors of a Factors of c Possible Factorizations Middle Term when multiplied
121
Examples 9.6

122
EXAMPLE 1
Factor when b is negative and c is positive
Factor 2x2 7x 3.
SOLUTION
Because b is negative and c is positive, both
factors of c must be negative. Make a table to
organize your work.
You must consider the order of the factors of 3,
because the x-terms of the possible
factorizations are different.
123
EXAMPLE 1
Factor when b is negative and c is positive
Factor 2x2 7x 3.
124
EXAMPLE 2
Factor when b is positive and c is negative
Factor 3n2 14n 5.
SOLUTION
Because b is positive and c is negative, the
factors of c have different signs.
125
EXAMPLE 2
Factor when b is negative and c is positive
126
for Examples 1 and 2
GUIDED PRACTICE
Factor the trinomial.
1. 3t2 8t 4.
SOLUTION
Because b is positive and c is positive, both
factors of c are positive.
You must consider the order of the factors of 4,
because the t-terms of the possible
factorizations are different.
127
for Examples 1 and 2
GUIDED PRACTICE
128
for Examples 1 and 2
GUIDED PRACTICE
Factor the trinomial.
2. 4s2 9s 5.
SOLUTION
Because b is negative and c is positive, both
factors of c must be negative. Make a table to
organize your work.
You must consider the order of the factors of 5,
because the s-terms of the possible
factorizations are different.
129
for Examples 1 and 2
GUIDED PRACTICE
130
for Examples 1 and 2
GUIDED PRACTICE
Factor the trinomial.
3. 2h2 13h 7.
SOLUTION
Because b is positive and c is negative, the
factors of c have different signs.
131
EXAMPLE 2
Factor when b is negative and c is positive
132
EXAMPLE 3
Factor when a is negative
Factor 4x2 12x 7.
SOLUTION
STEP 1
Factor 1 from each term of the trinomial.
4x2 12x 7 (4x2 12x 7)
STEP 2
Factor the trinomial 4x2 12x 7. Because b and
c are both negative, the factors of c must have
different signs. As in the previous examples, use
a table to organize information about the factors
of a and c.
133
EXAMPLE 3
Factor when a is negative
134
EXAMPLE 3
Factor when a is negative
135
for Example 3
GUIDED PRACTICE
Factor the trinomial.
4. 2y2 5y 3
SOLUTION
STEP 1
Factor 1 from each term of the trinomial.
2y2 5y 3 (2y2 5y 3)
STEP 2
Factor the trinomial 2y2 5y 3. Because b and
c are both positive, the factors of c must have
both positive. Use a table to organize
information about the factors of a and c.
136
for Example 3
GUIDED PRACTICE
137
for Example 3
GUIDED PRACTICE
Factor the trinomial.
5. 5m2 6m 1
SOLUTION
STEP 1
Factor 1 from each term of the trinomial.
5m2 6m 1 (5m2 6m 1)
STEP 2
Factor the trinomial 5m2 6m 1. Because b is
negative and c is positive, the factors of c must
be both negative. Use a table to organize
information about the factors of a and c.
138
for Example 3
GUIDED PRACTICE
139
EXAMPLE 4
Write and solve a polynomial equation
Discus
An athlete throws a discus from an initial height
of 6 feet and with an initial vertical velocity
of 46 feet per second.
140
EXAMPLE 4
Write and solve a polynomial equation
SOLUTION
a. Use the vertical motion model to write an
equation for the height h (in feet) of the
discus. In this case, v 46 and s 6.
h 16t2 vt s
Vertical motion model
h 16t2 46t 6
Substitute 46 for v and 6 for s.
b. To find the number of seconds that pass
before the discus lands, find the value of t for
which the height of the discus is 0. Substitute
0 for h and solve the equation for t.
141
EXAMPLE 4
Write and solve a polynomial equation
0 16t2 46t 6
Substitute 0 for h.
0 2(8t2 23t 3)
Factor out 2.
0 2(8t 1)(t 3)
Factor the trinomial. Find factors of 8 and 3
that produce a middle term with a coefficient of
23.
8t 1 0
or
t 3 0
Zero-product property
or
t 3
Solve for t.
142
EXAMPLE 4
Write and solve a polynomial equation
143
Warm-Up 9.7 and 9.8
144
Lesson 9.7, For use with pages 600-605
Find the product.
1. (m 2)(m 2)
2. (2y 3)2
145
Lesson 9.7, For use with pages 600-605
Find the product.
3. (s 2t)(s 2t)
146
Lesson 9.8, For use with pages 606-613
1. Solve 2x2 11x 21.
2. Factor 4x2 10x 4.
147
Lesson 9.8, For use with pages 606-613
148
Vocabulary 9.7 and 9.8

• Perfect Square Trinomial
• (a b)2 a2 2ab b2
• (a - b)2 a2 - 2ab b2
• Difference of two squares
• a2 b2 (a b)(a b)
• Factor by Grouping
• Look for this when you have 4 terms in a
  polynomial
• Factor out GCF from first two terms and second
  two terms.
• Factor Completely
• Polynomial w/ integer coefficients that cant be
  factored any more

149
Notes 9.7 and 9.8

• Before you factor a polynomial, FACTOR OUT THE
  GCF IN ALL THE TERMS FIRST!
• The GCF can be a polynomial as well!!
• Try these steps to ensure a polynomial is factored

150
Examples 9.7 and 9.8

151
EXAMPLE 1
Factor the difference of two squares
Factor the polynomial.
a. y2 16 y2 42
Write as a2 b2.
(y 4)(y 4)
Difference of two squares pattern
b. 25m2 36 (5m)2 62
Write as a2 b2.
(5m 6)(5m 6)
Difference of two squares pattern
c. x2 49y2 x2 (7y)2
Write as a2 b2.
(x 7y)(x 7y)
Difference of two squares pattern
152
EXAMPLE 2
Factor the difference of two squares
Factor the polynomial 8 18n2.
8 18n2 2(4 9n2)
Factor out common factor.
222 (3n) 2
Write 4 9n2 as a2 b2.
2(2 3n)(2 3n)
Difference of two squares pattern
153
for Examples 1 and 2
GUIDED PRACTICE
Factor the polynomial.
1. 4y2 64 (2y)2 (8)2
Write as a2 b2.
(2y 8)(2y 8)
Difference of two squares pattern
154
Factor perfect square trinomials
EXAMPLE 3
Factor the polynomial.
Write as a2 2ab b2.
(n 6)2
Perfect square trinomial pattern
Write as a2 2ab b2.
(3x 2)2
Perfect square trinomial pattern
Write as a2 2ab b2.
(2s t)2
Perfect square trinomial pattern
155
Factor a perfect square trinomial
EXAMPLE 4
Factor the polynomial 3y2 36y 108.
Factor out 3.
3y2 36y 108
3(y2 12y 36)
Write y2 12y 36 as a2 2ab b2.
3(y 6)2
Perfect square trinomial pattern
156
for Examples 3 and 4
GUIDED PRACTICE
Factor the polynomial.
2. h2 4h 4
Write as a2 2ab b2.
(h 2)2
Perfect square trinomial pattern
3. 2y2 20y 50
2(y2 10y 25)
Factor out 2
Write as y2 10y25 as a2 2abb2 .
2(y 5)2
Perfect square trinomial pattern
157
for Examples 3 and 4
GUIDED PRACTICE
4. 3x2 6xy 3y2
3(x2 2xy y2)
Factor out 3
Write as x2 2xy y2 as a2 2abb2 .
3(x y)2
Perfect square trinomial pattern
158
EXAMPLE 1
Factor out a common binomial
Factor the expression.
SOLUTION
3y2(y 2) 5(2 y) 3y2(y 2) 5(y 2)
Factor 1 from (2 y).
(y 2)(3y2 5)
Distributive property
159
EXAMPLE 2
Factor by grouping
Factor the polynomial.
SOLUTION
Group terms.
x2(x 3) 5(x 3)
Factor each group.
(x 3)(x2 5)
Distributive property
Group terms.
y(y 1) x(y 1)
Factor each group.
(y 1)(y x)
Distributive property
160
EXAMPLE 3
Factor by grouping
SOLUTION

x3 3x2 2x 6
Rearrange terms.

Group terms.
Factor each group.
Distributive property
161

EXAMPLE 3
Factor by grouping
CHECK
162
for Examples 1, 2 and 3
GUIDED PRACTICE
Factor the expression.
1. x (x 2) (x 2)
x (x 2) (x 2)
x (x 2) 1(x 2)
Factor 1 from x 2.
(x 2) (x 1)
Distributive property
2. a3 3a2 a 3.
(a3 3a2) (a 3)
a3 3a2 a 3
Group terms.
a2(a 3) 1(a 3)
Factor each group.
(a2 1)(a 3)
Distributive property
163
for Examples 1, 2 and 3
GUIDED PRACTICE
3. y2 2x yx 2y.
SOLUTION
The terms y2 and 2x have no common factor. Use
the commutative property to rearrange the terms
so that you can group terms with a common factor.
y2 2x yx 2y
y2 yx 2y 2x
Rearrange terms.
( y2 yx ) ( 2y 2x )
Group terms.
y( y x ) 2(y x )
Factor each group.
(y 2)( y x )
Distributive property
164
Review Ch. 9 PUT HW QUIZZES HERE
165

Daily Homework Quiz
For use after Lesson 9.1
If the expression is a polynomial, find its
degree and classify it by the number of
terms.Otherwise, tell why it is not a polynomial
1. m3 n4m2 m2
2. 3b3c4 4b2cc8
166

Daily Homework Quiz
For use after Lesson 9.1
Find the sum or difference.
3. (3m2 2m9) (m22m 4)
4. ( 4a2 3a 1) (a2 2a 6)
167

Daily Homework Quiz
For use after Lesson 9.1
5. The number of dog adoptions D and cat
adoptions C can be modeled by
D 1.35 t2 9.8t131 and C 0.1t2 3t79 where
t represents the years since 1998. About how many
dogs and cats were adopted in 2004?
168

Daily Homework Quiz
For use after Lesson 9.2
Find the product.
1. 3x(x3 3x2 2x 4)
2. (y 4)(2y 5)
169

Daily Homework Quiz
For use after Lesson 9.2
3. (4x 3)(3x 2)
4. (b2 2b 1)(3b 5)
170

Daily Homework Quiz
For use after Lesson 9.2
5. The dimensions of a rectangle are x4 and
3x 1.Write an expression to represent the area
of the rectangle.
171

Daily Homework Quiz
For use after Lesson 9.3
Find the product.
1. (y 8)(y 8)
2. (3m 2n)2
172

Daily Homework Quiz
For use after Lesson 9.3
3. (2m 5)2
4. In humans, the genes for being able to roll
and not roll the tongue and R and r,
respectively. Offspring with R can roll the
tongue. If one parent is Rr and the other is
rr,what percent of the offspring will not be able
to roll the tongue?
173
Daily Homework Quiz
For use after Lesson 9.4
Solve the equation.
1. (y 5 ) (y 9 ) 0
2. (2n 3 ) (n 4 ) 0
3. 6x2 20x
174
Daily Homework Quiz
For use after Lesson 9.4
4. 12x2 18x
175
Daily Homework Quiz
For use after Lesson 9.5
Factor the trinomial.
1. x2 6x 16
2. y2 11y 24
3. x2 x 12
176
Daily Homework Quiz
For use after Lesson 9.5
4. Solve a2 a 20
177
Daily Homework Quiz
For use after Lesson 9.6
Factor the trinomial.
1. x2 x 30
2. 5b2 3b 14
2. 6y2 13y 5
178
Daily Homework Quiz
For use after Lesson 9.6
4. Solve 2x2 7x 3
179
Daily Homework Quiz
For use after Lesson 9.7
Factor the trinomial.
1. 4m2 n2
2. x2 6x 9
3. 4y2 16y 16
180
Daily Homework Quiz
For use after Lesson 9.7
4. Solve the equation
181

Daily Homework Quiz
For use after Lesson 9.8
Factor the polynomial completely.
1. 40b5 5b3
2. x3 6x2 7x
3. y3 6y2 y 6
182

Daily Homework Quiz
For use after Lesson 9.8
4. Solve 2x3 18x2 40x.
5. A sewing kit has a volume of 72 cubic
inches.Its dimensions are w,w 1, and 9 w
units.Find the dimensions of the kit.