Chapter 8: Factoring. Greatest Common Factor (GCF) 8.1 Grouping 8.2 Trinomials – x 2 + bx + c 8.3 Trinomials – ax 2 + bx + c 8.4 Differences of Squares.

Chapter 8: Factoring

Chapter 8 : Factoring

Greatest Common Factor (GCF) 8.1

Grouping 8.2

Trinomials – x2 + bx + c 8.3

Trinomials – ax2 + bx + c 8.4

Differences of Squares 8.5

Perfect Squares 8.6

Sums and Differences of Cubes

Quadratic Form

Combinations

Fill in the titles on the foldable

8.2 Greatest Common Factor (top) Find the GCF of the terms Write the GCF then the remaining part of

each term in parentheses

8.2 Greatest Common Factor (bottom)

Ex: 12a2 + 16a Ex: 3p2q – 9pq2 + 36pq

2 x 2 x 3 x a x a

2 x 2 x 2 x 2 x a= 2 x 2 x a =4a

4a(3a + 4)

x 3 x p x p x q-1 x 3 x 3 x p x q x q2 x 2 x 3 x 3 x p x q

3 x p x q = 3pq

3pq(p - 3q + 12)

Your Turn – What is the Greatest Common Factor (GCF)

Ex: 15x + 20 Ex: 8x³y + 2x²y² - 4 xy³

GCF=5

=5(3x + 4)

2 x x x y = 2xy

=2xy(4x² + xy – 2y²)

8.2 Factor by Grouping (top) Group the terms (first two and last two) Find the GCF of each group Write each group as a product of the GCF

and the remaining factors Combine the GCFs in a group and write the

other group as the second factor

8.2 Factor by Grouping (bottom)

Ex: 4ab + 8b + 3a + 6

(4ab + 8b)( +3a + 6)

2 x 2 x a x b

2 x 2 x 2 x b=4b

4b(a + 2)

3 x a

2 x 3 = 3

+3 (a + 2)

(4b + 3)(a + 2)

Ex: 3p – 2p2 – 18p + 27

(3p – 2p2 )( – 18p + 27)

3 x p

-1 2 x p x p= p -1 x 2 x 3 x 3 x p

3 x 3 x 3 = 9

p(3 – 2p) + 9(-2p + 3)

(p + 9)(-2p + 3)

Your turn- Factor by Grouping

Ex: 5x(x – 2) + 6(x – 2)

(x – 2) (5x + 6)

Ex: 3x² - 2x + 6x – 4

(3x² – 2x )+( 6x - 4)

x(3x – 2) + 2(3x - 2)

(x + 2)(3x – 2)

Factoring Trinomials – x2 + bx + c

• Get everything on one side (equal to zero)

• Split into two groups ( )( ) = 0

• Factor the first part x2 (x )(x ) = 0

• Find all the factors of the third part (part c)

• Fill in the factors of c that will add or subtract to make the second part (bx)

• Use Distributive Property (Foil) to check your answer

• Use Zero Product Property to solve if needed

Factoring Trinomials – x2 + bx + c

Ex: x2 + 6x + 8

(x )(x ) 8

1, 8

2, 4(x + 2)(x + 4)

FOIL

x2 + 2x + 4x + 8

x2 + 6x + 8

Ex: r2 – 2r - 24

(r )(r ) 24

1, 24

2, 12

3, 8

4, 6

(r + 4)(r - 6)

FOIL

r2 – 6r + 4r - 24

r2 - 2x - 24

Ex: s2 – 11s + 28 = 0

(s )(s ) 28

1, 28

2, 14

4, 7

(s- 4)(s - 7) = 0

FOIL

s2 – 7s – 4s + 28

s2 – 11s + 28

s – 4 = 0 s – 7 = 0 +4 +4 +7 +7

s = 4 s = 7

s = 4 and 7

Check your work Check your workCheck your work

8.4 Factoring Trinomials – ax2 + bx + c (top) Get everything on one side (equal to zero) Find product of the first and last parts Find the factors of the product Rewrite the ax2 then fill in the pair of factors

that adds or subtracts to make the second part followed by c

Factor by grouping

if you can’t factor = prime(use the zero product property to solve if needed)

8.4 Factoring Trinomials – ax2 + bx + c (bottom)

Ex: 5x2 + 13x + 6 Ex: 10y2 - 35y + 30 = 05 x 6 = 30

1, 30

2, 15

3, 10

5, 6

x(5x + 3) + 2(5x + 3)

(x + 2)(5x + 3)

5(2y2 - 7y + 6) = 0

Hint: find the gcf to pull it out and make the numbers smaller if possible

2 x 6 = 12

1, 12

2, 6

3, 4

5(y - 2)(2y - 3) = 0

y – 2 = 0 2y – 3 = 0Solve for y.

y = 2 and 1.5

5x2 + 3x + 10x + 65[(2y2 -3y)(-4y + 6)]=05[y(2y - 3)-2(2y - 3)]=0

( ) ( )

8.5 Factoring Differences of Squares (top) Factor each term Write one set of parentheses with the factors adding

and one with the factors subtracting Foil to check your answer

Ex: n2 - 25

n x n 5 x 5

(n + 5)(n - 5)

8.5 Factoring Differences of Squares (bottom)

Ex: 5x3 + 15x2 – 5x - 15 Ex: 121a = 49a3

5[x3 + 3x2 – x – 3]

5[ (x3 + 3x2)( – x – 3)]

5[ x2(x + 3) - 1(x + 3)]

5[(x2 – 1)(x + 3)]

5[(x x x 1 x 1)(x + 3)]

5(x + 1)(x - 1)(x + 3)

-121a -121a

0 = 49a3 – 121a0 = a(49a2 – 121)

0 = a(7a x 7a 11 x 11)

0 = a(7a + 11)(7a - 11)

a = 0 7a + 11 = 0 7a - 11 = 0 -11 -11 +11 +117a = -11 7a = 11/7 /7 /7 /7

a= -11/7 a = 11/7

a = -11/7, 0, and 11/7

8.6 Factoring Perfect Squares (top) Perfect Square Trinomial:

Is the first term a perfect square? Is the last term a perfect square? Does the second term = 2 x the product of the

roots of the first and last terms?

The third term (c) must be positive Use the sign of the second term If any of these answers is no- it is not a perfect

square trinomial

8.6 Factoring Perfect Squares (bottom)

Ex: Ex:x2 – 14x + 49 a2 – 8a - 16

Ex: 9y2 + 12y + 4

1. 9y2 = 3y x 3y yes

2. 4 = 2 x 2 yes

3. 2(3y x 2) = 2(6y) = 12y yes

(3y + 2)2

(x – 7)2

7 x 7x x x2 x x x 7= 14x

4 x 4a x a

4 x 4 = 16 but it is a negative 16 so it can’t be a perfect square

Sum and Difference of Cubes(top)

a3 + b3 = (a + b)(a2 – ab + b2)

a3 - b3 = (a - b)(a2 + ab + b2)

Same - Opposite - Always Positive

To remember the signs:

SOAP

Sum and Difference of Cubes(bottom)Ex: x3 + 8 Ex: 27x3 – 64y3

Ex: 1000y3 – 216 Ex: 125a3 + 27b3

(x + 2) (x2 – 2x + 4)

x x x 2 2 2

x2 - 1x2x + 22

3x 3x 3x 4y 4y 4y

(3x - 4y) (9x2 + 12xy + 16y2)

(10y - 6) (100x2 + 60y + 36) (5a + 3b) (25a2 – 15ab + 9b2)

Quadratic Form(Bottom)Ex: x4 + 3x2 + 2 Ex: x4 – 16

Ex: x - + 21 Ex: x4 + 7x2 + 12x10

(x2 )(x2 )

(x2 + 1)(x2 + 2)

(x2 )(x2 )

(x2 + 1)(x2 + 2)(x2 – 2)

(x2 + 4)(x2 - 4)

( - 3)( - 7)x x (x2 )(x2 )

(x2 + 3)(x2 + 4)

Combinations of Factoring Types(Top)

First look for the GCF and factor out if possible

Next look for patterns (perfect squares, difference of squares or sum and difference of cubes)

If no patterns appear factor the trinomial like normal

Combinations of Factoring Types(bottom)Ex: 4x2 – 100 Ex: 3x2 – 3x – 60

Ex: 8x6 – 64x3 Ex: 8x3 – 32x

4(x2 – 25)

4(x + 5)(x – 5)

3(x2 – x - 20) 20

1, 20

2, 10

4, 5

(x + 4)(x – 5)

8x3 (x3 – 8)

8x3(x – 2)(x2 + 2x + 4)

8x(x2 – 4)

8x(x + 2)(x – 2)