Factoring Expressions

This topic will be used throughout the 2nd semester.

Weekly Quizzes

What is Factoring?• Quick Write: In your notes, write down everything

you know about Factoring from Algebra-1 and Geometry.

• You can use Bullets or give examples

• 2 Minutes

Objectives• I can factor expressions using all of these

methods– Greatest Common Factor (GCF)– Difference of Two Squares– Reverse FOIL– Swing and Divide

• I can solve a quadratic equation using factoring

Factoring?• Factors are two or more terms that

multiply to form a product• (Factor) x (Factor) = Product• Some numbers are Prime, meaning

they are only divisible by themselves and 1

Method 1• Greatest Common Factor (GCF) –

the greatest factor shared by two or more terms

• ALWAYS try this factoring method 1st before any other method

• Divide Out the Biggest common number/variable from each of the terms

Greatest Common Factorsaka GCF’s

Find the GCF for each set of following numbers.Find means tell what the terms have in common.Hint: list the factors and find the greatest match.

a) 2, 6b) -25, -40c) 6, 18d) 16, 32e) 3, 8

2-56

161

No common factors? GCF =1

Find the GCF for each set of following numbers.

Hint: list the factors and find the greatest match.

a) x, x2

b) x2, x3

c) xy, x2yd) 2x3, 8x2

e) 3x3, 6x2

f) 4x2, 5y3

xx2

xy2x2

Greatest Common Factorsaka GCF’s

3x2

1 No common factors? GCF =1

Factor out the GCF for each polynomial:Factor out means you need the GCF times the

remaining parts.

a) 2x + 4yb) 5a – 5bc) 18x – 6yd) 2m + 6mne) 5x2y – 10xy

2(x + 2y)

6(3x – y)

5(a – b)

5xy(x - 2)

2m(1 + 3n)

Greatest Common Factorsaka GCF’s

How can you check?

Ex 1

•15x2 – 5x•GCF = 5x•5x(3x - 1)

Ex 2

•8x2 – x•GCF = x•x(8x - 1)

Method #2

•Difference of Two Squares•a2 – b2 = (a + b)(a - b)

What is a Perfect Square

• Any term you can take the square root evenly (No decimal)

• 25• 36• 1• x2

• y4

5

6

1x

2y

Difference of Perfect Squares

x2 – 4 =

the answer will look like this: ( )( )

take the square root of each part:( x 2)(x 2)

Make 1 a plus and 1 a minus:(x + 2)(x - 2 )

FACTORINGDifference of PerfectSquares

EX:x2 – 64

How:Take the square root of each part. One gets a + and one gets a -.Check answer by FOIL.

Solution:(x – 8)(x + 8)

Example 1

•(9x2 – 16)•(3x + 4)(3x – 4)

Example 2

•x2 – 16•(x + 4)(x –4)

Ex 3

•36x2 – 25•(6x + 5)(6x – 5)

Ex 4

•9x2 + 25•PRIME

FOIL Review

( 7)( 3)x x First Terms Outside Terms Inside Terms Last Terms

2x 3x 7x 21

2 4 21x x

Reverse FOIL

• When factoring a trinomial with 3 terms you will always get two factors

• x2 + bx + c

( #)( *)x x

Factoring a Trinomial

2x bx c ( ___)( ___)x x

What do these numbers have to be?

2x bx c ( ___)( ___)x x

What do these numbers have to be?They MULTIPLY to "c"

They ADD to "b"

FOIL Method

( #)( *)x x

2 5 6x x

The two numbers # and * must multiply together to equal 6The two numbers # and * must add up to 5

Example 1

2 5 6x x

( __)( __)x x

Factoring Tree6

1 61 6 2 32 3

( 2)( 3)x x

Example 2

2 3 18x x

( __)( __)x x

Factoring Tree18

1 181 182 92 9

( 3)( 6)x x

3 63 6

Example 3

2 3 8x x

( __)( __)x x

Factoring Tree

81 81 82 4

2 4PRIME

Swing and Divide Method

• The Swing & Divide method is very similar to Reverse FOIL, but with 2 extra steps:

• You can use this method when the number in front of the x2 term is not 1

• EX: 2x2 – x - 3

Swing & Divide

• 2 Steps: • Swing and • Divide• Don’t do one without doing

the 2nd

Swing & Divide Method

22 6x x

Swing

2 12x x Now use FOIL

( 4)( 3)x x

4 3( )( )2 2

x x Divide by Swing #

Final Factors

( 2)(2 3)x x

Reducing and Checking

• Always reduce fractions before finding final factors

• Use TWO Finger Check when done

Example 2

23 7 6x x

Swing

2 7 18x x Now use FOIL

( 9)( 2)x x

9 2( )( )3 3

x x Divide by Swing #

Final Factors

( 3)(3 2)x x

Example 3

28 2 3x x

Swing

2 2 24x x Now use FOIL

( 6)( 4)x x

6 4( )( )8 8

x x Divide by Swing #

Final Factors

(4 3)(2 1)x x

3 1( )( )4 2

x x Reduce

Board 122 5 3x x

2 5 6x x ( 6)( 1)x x

6 1( )( )2 2

x x

( 3)(2 1)x x

Board 222 3 9x x

2 3 18x x ( 6)( 3)x x

6 3( )( )2 2

x x

( 3)(2 3)x x

Board 329 12 4x x

2 12 36x x ( 6)( 6)x x

6 6( )( )9 9

x x

(3 2)(3 2)x x

More than ONE Method

• It is very possible to use more than one factoring method in a problem

• Remember:

• ALWAYS use GCF first

Example 1

• 2b2x – 50x• GCF = 2x• 2x(b2 – 25) • 2nd term is the diff of 2 squares• 2x(b + 5)(b - 5)

Example 2

• 32x3 – 2x• GCF = 2x• 2x(16x2 – 1) • 2nd term is the diff of 2 squares• 2x(4x + 1)(4x - 1)

Equations

• To solve an equation by factoring, the equation MUST be set equal to ZERO first:

ax2 + bx + c = 0

Zero Product Property

•If ab = 0, then•a = 0• b = 0

Solving with Factoring

• Set each variable factor equal to zero and solve for “x”

(Factor #1)(Factor #2) = 0ThenFactor #1 = 0 and Factor #2 = 0

Factoring for solutions• x2 – 3x –4 = 0• (x – 4)(x + 1) = 0• Then by the Zero Product Property:• (x – 4) = 0 or (x + 1) = 0 • If we solve these for “x” we get the following

solutions:• x – 4 = 0, so x = 4• x + 1 = 0, so x = -1• These are the two solutions {-1, 4}

Example 222 6 0x x

(2 3)( 2) 0x x

(2 3) 0 ( 2) 0x x

3 22

x x

Example 3

210 2 0x x

2 (5 1) 0x x

2 0 (5 1) 0x x

105

x x

Example 422 6 36 0x x

2( 6)( 3) 0x x

( 6) 0 ( 3) 0x x

6 3x x

22( 3 18) 0x x

Homework

• WS 5-1