Algebra 3

Warm-Up 2.2

List the factors of 36

1, 362, 183, 124, 9

6,

Algebra 3

Lesson 2.2

Objective:

SSBAT factor a polynomial by factoring out the GCF and using Difference of Squares.

Standards: M11.D.2.2.2

Monomial An expression with 1 term

5x2 -3mn3k8

BinomialAn expression with 2 terms

4x – 215x3 + 8y

TrinomialAn expression with 3 terms

8x5 – 5w3 + 216 – 3x + 5m4

Factors

The numbers used in a multiplication problem

5 x 3 = 15, 5 and 3 are the factors of 15

List the Factors of 24

1, 2, 3, 4, 6, 8, 12, 24

Greatest Common Factor (GCF)

The biggest number that is a factor of all of the numbers in a set.

Find the GCF of 18 and 45

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 45: 1, 3, 5, 9, 15, 45

GCF of 18 and 45 is 9

Finding the GCF of expressions (variables)Example: 6x4y and 10x2

1. Find the GCF of the Coefficients (numbers in front)

2. Find the GCF of each variable piece by

Look at only one set of like variables at a time

If the variable does not appear in all of the terms do not use it in the GCF

If the variable appears in every term use the one with the smaller exponent in the GCF

Find the GCF of each.

1. 25x2y4 and 10x3y

GCF: 5x2y

2. 21mn2k and 10m2n2

mn2

3. 12x4y, 9x5y2, 21x7yw3

GCF: 3x4y

4. m5n3k2 and mnk

GCF: mnk

Factoring

Rewriting an expression as a multiplication problem

2 · 5 is the factored form of 10

3(x + 4) is the factored form of 3x + 12

Factoring Out the GCF

1. Find the GCF of all of the terms in the polynomial

2. Write the GCF outside of the parentheses

3. Divide each term of the polynomial by the GCF and write this expression inside the parentheses

Examples: Factor out the GCF of each.

1. 2w3 + 10w

The GCF is 2w

= 2w(w2 + 5)

Examples: Factor out the GCF of each.

2. 18n3 + 9n2 – 24n

The GCF is 3n

= 3n(6n2 + 3n – 8)

Examples: Factor out the GCF of each.

3. –20x6 + 12x3 – 4x

The GCF is 4x When the lead coefficient is Negative factor out the negative as well

= –4x(5x5 – 3x2 + 1)

Examples: Factor out the GCF of each.

4. 15mn3 – 9m2n4 + 18m3n5

The GCF is 3mn3

= 3mn3(5 – 3mn + 6m2n2)

Perfect Square

A number that you can take the Square Root of

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,…

Perfect Square expressions are x2, x4, x6, …

Difference of Squares

An expression of the a2 – b2

Perfect Square – Perfect Square

Examples: x2 – 4

m2 – 25

9x2 – 1

*** x2 + 4 is NOT a difference of squares because of the PLUS ***

Factoring a Difference of Squares

a2 – b2 = (a + b)(a – b)

Take the square root of each term

Add the 2 square roots in one ( )

Subtract the 2 square roots in another ( )

Factor each Difference of Squares

1. x2 – 64

= ( + )( – )

2. 81 – x2

=( + )( – )

x x8 8

9 9x x

Factor each Difference of Squares

3. 4x2 – 25

= ( + )( – )

4. w2 – y2

= ( + )( – )

2x 2x5 5

w wy y

Factor each Difference of Squares

5. 49x2 – 1

= ( + )( – )

6. x2 – 130

Can’t Do – It is NOT a Difference of Squares 130 is not a perfect square

7. x2 + 9

Can’t Do – It is NOT a Difference of Squares It’s Plus not Minus

7x 7x1 1

8. w6 – 196

= (w3 + 14)(w3 – 14)

9. 100x22 – y16

= (10x11 – y8)(10x11 + y8)

On Your Own.

1. Factor out the GCF. 8x3 – 20x5 + 2x2

2. Factor the Difference of Squares. 36x2 – 49

3. Factor out the GCF. 12m3n5 – 24mn4 – 30m6n

4. Factor the Difference of Squares. 81 – m12

= 2x2(4x – 10x3 + 1)

= (6x + 7)(6x – 7)

= (9 + m6)(9 – m6)

= 6mn(2m2n4 – 4n3 – 5m5)

Homework

Worksheet 2.2