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Warm-Up: September 22, 2015Simplify

Homework Questions?

Factoring Polynomials

Section P.5

Factoring is the process of writing a polynomial as the product of two or more polynomials.

Prime polynomials cannot be factored using integer coefficients.

Factor completely means keep factoring until everything is prime

Factoring

Greatest common factor Difference of two perfect squares Perfect-square trinomials Factoring x2 + bx + c (big X) Factoring ax2 + bx + c (big X) Factor by grouping – use with 4 terms Sum and difference of perfect cubes

Methods of Factoring

Find the greatest common factor (GCF) of all terms.

Divide each term by the greatest common factor.

Write the GCF outside parenthesis, with the rest of the divided terms added together inside

3a2 – 12a 3a is the GCF

Factoring out greatest common factor

4312,

33 2

aaa

aa

43 aa

a. 18x3 + 27x2 b. x2(x + 3) + 5(x + 3)

You-Try #1: Factor

Works with an even number of terms. Split the terms into two groups. Factor each group separately using GCF. If factor by grouping is possible, the part

inside parentheses of each group will be the same.

Treat the parentheses as common factors to finish factoring.

Factor by Grouping

1535 23 xxxExample 2: Factor

1226 23 xxxYou-Try #2: Factor

Factoring x2 + bx + c Look for integers r and s such that:

◦ r × s = c◦ r + s = b

sxrxcbxx 2

c

b

r s

Example 31272 xx

You-Try #3652 xx

You-Try #3652 xx

Factoring ax2 + bx + c Look for integers r and s such that:

◦ r × s = ac◦ r + s = b

Divide r and s by a, then reduce fractions In your factors, any remaining denominator

gets moved in front of the x

ac

b

r s

Example 46135 2 xx

You-Try #43116 2 xx

Factoring Perfect Square Trinomials

Let A and B be real numbers, variables, or algebraic expressions, 1. A2 + 2AB + B2 = (A + B)2

2. A2 – 2AB + B2 = (A – B)2

Factor: 16x2 – 56x + 49

Example 7

Factor: x2 + 14x + 49

You-Try #7

bababa 22

yxyxyx

xxx

xxx

75754925

121214

339

22

2

2

Difference of Two Perfect Squares

Factoring the Sum and Difference of 2 Cubes

64x3 – 125 = (4x)3 – 53 = (4x – 5)((4x)2 + (4x)(5) + 52) = (4x – 5)(16x2 + 20x + 25)

A3 – B3 = (A – B)(A2 + AB + B2)

x3 + 8 = x3 + 23 = (x + 2)( x2 – x·2 + 22) = (x + 2)( x2 – 2x + 4)

A3 + B3 = (A + B)(A2 – AB + B2)ExampleType

2233 BABABABA

33 8125 yx

Example 8

100027 3 x

You-Try #8

1. If there is a common factor, factor out the GCF.2. Determine the number of terms in the

polynomial and try factoring as follows:a) If there are two terms, can the binomial be factored

by one of the special forms including difference of two squares, sum of two cubes, or difference of two cubes?

b) If there are three terms, is the trinomial a perfect square trinomial? If the trinomial is not a perfect square trinomial, try factoring using the big X.

c) If there are four or more terms, try factoring by grouping.

3. Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely.

A Strategy for Factoring a Polynomial

Factoring FlowchartFactor out GCF

Count number of terms2

3

4

Factor byGrouping

Check for:• Difference of perfect squares• Sum of perfect cubes• Difference of perfect cubes

1. Check for perfect square trinomial

2. Use big X factoring

Check each factor to see if it can be factored further

Page 53 #1-75 Odd

Assignment

In Exercises 1-10, factor out the greatest common factor.

In Exercises 11-16, factor by grouping.

In Exercises 17-30, factor each trinomial, or state that the trinomial is prime.

158)21

152)19

65)17

4623)15

22)13

1052)11

3123)9

535)727189)5

63)3

2718)1

2

2

2

23

23

23

2

234

2

xx

xx

xx

xxx

xxx

xxx

xxx

xxxxxx

xx

x

In Exercises 17-30, factor each trinomial, or state that the trinomial is prime.

In Exercises 31-40, factor the difference of two squares.

8116)39

16)37

259)35

4936)33

100)31

15164)29

4116)27

28253)25

23)23

4

4

22

2

2

2

2

2

2

x

x

yx

x

x

xx

xx

xx

xx

In Exercises 41-48, factor any perfect square trinomials, or state that the polynomial is prime.

In Exercises 49-56, factor using the formula for the sum or difference of two cubes.

In Exercises 57-84, factor completely, or state that the polynomial is prime.

1622)61

2444)59

33)57

2764)55

18)53

64)51

27)49

169)47

144)45

4914)43

12)41

4

2

3

3

3

3

3

2

2

2

2

x

xx

xx

x

x

x

x

xx

xx

xx

xx