Algebra unit 8.8

UNIT 8.7 FACTORING SPECIAL CASESUNIT 8.7 FACTORING SPECIAL CASES

Warm Up

Factor each trinomial.

1. x2 + 13x + 40

2. 5x2 – 18x – 8

3. Factor the perfect-square trinomial 16x2 + 40x + 25

4. Factor 9x2 – 25y2 using the difference of two squares.

(x + 5)(x + 8)

(4x + 5)(4x + 5)

(5x + 2)(x – 4)

(3x + 5y)(3x – 5y)

Choose an appropriate method for factoring a polynomial.

Combine methods for factoring a polynomial.

Objectives

Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.

Example 1: Determining Whether a Polynomial is Completely Factored

Tell whether each polynomial is completely factored. If not factor it.

A. 3x2(6x – 4)

3x2(6x – 4)

Neither x2 +1 nor x – 5 can be factored further.

6x2(3x – 2)

B. (x2 + 1)(x – 5) (x2 + 1)(x – 5)

6x – 4 can be further factored.

Factor out 2, the GCF of 6x and – 4.

6x2(3x – 2) is completely factored.

(x2 + 1)(x – 5) is completely factored.

x2 + 4 is a sum of squares, and cannot be factored.

Caution

Check It Out! Example 1

Tell whether the polynomial is completely factored. If not, factor it.

A. 5x2(x – 1)

5x2(x – 1) Neither 5x2 nor x – 1 can be factored further.

B. (4x + 4)(x + 1)

Factor out 4, the GCF of 4x and 4.

4x + 4 can be further factored.

5x2(x – 1) is completely factored.

4(x + 1)2 is completely factored.

(4x + 4)(x + 1)

4(x + 1)(x + 1)

To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely.

Example 2A: Factoring by GCF and Recognizing Patterns

Factor 10x2 + 48x + 32 completely. Check your answer.

10x2 + 48x + 32

2(5x2 + 24x + 16)

2(5x + 4)(x + 4)

Factor out the GCF.

Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16)

= 10x2 + 40x + 8x + 32

= 10x2 + 48x + 32

Factor remaining trinomial.

Example 2B: Factoring by GCF and Recognizing Patterns

Factor 8x6y2 – 18x2y2 completely. Check your answer.

8x6y2 – 18x2y2

2x2y2(4x4 – 9)

Factor out the GCF. 4x4 – 9 is a perfect-square trinomial of the form a2 – b2.

2x2y2(2x2 – 3)(2x2 + 3) a = 2x, b = 3

Check 2x2y2(2x2 – 3)(2x2 + 3) = 2x2y2(4x4 – 9)

= 8x6y2 – 18x2y2

Check It Out! Example 2a

Factor each polynomial completely. Check your answer.

4x3 + 16x2 + 16x4x3 + 16x2 + 16x4x(x2 + 4x + 4)4x(x + 2)2

Factor out the GCF. x2 + 4x + 4 is a perfect-square trinomial of the form a2 + 2ab + b2.

a = x, b = 2

Check 4x(x + 2)2 = 4x(x2 + 2x + 2x + 4)

= 4x(x2 + 4x + 4)

= 4x3 + 16x2 + 16x

Check It Out! Example 2b

Factor each polynomial completely. Check your answer.

2x2y – 2y3

2y(x + y)(x – y)

Factor out the GCF. 2y(x2 – y2) is a perfect-square trinomial of the form a2 – b2.

a = x, b = y

Check 2y(x + y)(x – y) = 2y(x2 + xy – xy – y2)

= 2x2y – 2y3

2x2y – 2y3

2y(x2 – y2)

= 2x2y +2xy2 – 2xy2 – 2y3

If none of the factoring methods work, the polynomial is said to be unfactorable.

For a polynomial of the form ax2 + bx + c, if there are no numbers whose sum is b and whose product is ac, then the polynomial is unfactorable.

Helpful Hint

Example 3A: Factoring by Multiple Methods

Factor each polynomial completely.

9x2 + 3x – 2 9x2 + 3x – 2 ( x + )( x + )

The GCF is 1 and there is no pattern.

a = 9 and c = –2; Outer + Inner = 3

Factors of 9 Factors of 2 Outer + Inner

1 and 9 1 and –2 1(–2) + 1(9) = 7

3 and 3 1 and –2 3(–2) + 1(3) = –3 3 and 3 –1 and 2 3(2) + 3(–1) = 3

(3x – 1)(3x + 2)

Example 3B: Factoring by Multiple Methods


12b3 + 48b2 + 48b

(x + )(x + )

The GCF is 12b; (b2 + 4b + 4) is a perfect-square trinomial in the form of a2 + 2ab + b2.

a = 2 and c = 2

12b(b2 + 4b + 4)

12b(b + 2)(b + 2)

12b(b + 2)2

Factors of 4 Sum

1 and 4 52 and 2 4

Example 3C: Factoring by Multiple Methods


4y2 + 12y – 72 4(y2 + 3y – 18)

Factor out the GCF. There is no pattern. b = 3 and c = –18; look for factors of –18 whose sum is 3. (y + )(y + )

Factors of –18 Sum

–1 and 18 17 –2 and 9 7

–3 and 6 3

4(y – 3)(y + 6)

The factors needed are –3 and 6

Example 3D: Factoring by Multiple Methods.


(x4 – x2)

x2(x2 – 1) Factor out the GCF.

x2(x + 1)(x – 1) x2 – 1 is a difference of two squares.

Check It Out! Example 3a


3x2 + 7x + 43x2 + 7x + 4

( x + )( x + )

a = 3 and c = 4; Outer + Inner = 7


3 and 1 1 and 4 3(4) + 1(1) = 13

3 and 1 2 and 2 3(2) + 1(2) = 8 3 and 1 4 and 1 3(1) + 1(4) = 7

(3x + 4)(x + 1)

Check It Out! Example 3b


2p5 + 10p4 – 12p3

2p3(p2 + 5p – 6) Factor out the GCF. There is no pattern. b = 5 and c = –6; look for factors of –6 whose sum is 5.

(p + )(p + )

Factors of – 6 Sum

– 1 and 6 5

2p3(p + 6)(p – 1)

The factors needed are –1 and 6

Check It Out! Example 3c


Factor out the GCF. There is no pattern.3q4(3q2 + 10q + 8)

9q6 + 30q5 + 24q4

a = 3 and c = 8; Outer + Inner = 10

( q + )( q + )


3 and 1 1 and 8 3(8) + 1(1) = 25

3 and 1 2 and 4 3(4) + 1(2) = 14 3 and 1 4 and 2 3(2) + 1(4) = 10

3q4(3q + 4)(q + 2)

Check It Out! Example 3d


2x4 + 18

2(x4 + 9) Factor out the GFC.

x4 + 9 is the sum of squares and that is not factorable.

2(x4 + 9) is completely factored.

Tell whether the polynomial is completely factored. If not, factor it.1. (x + 3)(5x + 10) 2. 3x2(x2 + 9)

Lesson Quiz

(x + 4)(x2 + 3)

completely factoredno; 5(x+ 3)(x + 2)

4(x + 6)(x – 2)

5. 18x2 – 3x – 3 6. 18x2 – 50y2

7. 5x – 20x3 + 7 – 28x2

3(3x + 1)(2x – 1) 2(3x + 5y)(3x – 5y) (1 + 2x)(1 – 2x)(5x + 7)

Factor each polynomial completely. Check your answer. 3. x3 + 4x2 + 3x + 12 4. 4x2 + 16x – 48

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Education

Algebra unit 8.8