1. Choosing a Factoring Method
7-6
Choosing a Factoring Method
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. Objectives Choose an appropriate method for factoring a polynomial.
Combine methods for factoring a polynomial.

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

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

5. Example 2A: Factoring by GCF and Recognizing Patterns
Factor 10x2 + 48x + 32 completely. Check your answer.
10x2 + 48x + 32
2(5x2 + 24x + 16)
Factor out the GCF.
2(5x + 4)(x + 4)
Factor remaining trinomial.
Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16)
= 10x2 + 40x + 8x + 32
= 10x2 + 48x + 32 

6. Example 2B: Factoring by GCF and Recognizing Patterns
Factor 8x6y2 – 18x2y2 completely. Check your answer.
8x6y2 – 18x2y2
Factor out the GCF. 4x4 – 9 is a perfect-square trinomial of the form a2 – b2.
2x2y2(4x4 – 9)
2x2y2(2x2 – 3)(2x2 + 3)
a = 2x, b = 3
Check 2x2y2(2x2 – 3)(2x2 + 3) = 2x2y2(4x4 – 9)
= 8x6y2 – 18x2y2 

7. Check It Out! Example 2a
Factor each polynomial completely. Check your answer.
4x3 + 16x2 + 16x
Factor out the GCF. x2 + 4x + 4 is a perfect-square trinomial of the form a2 + 2ab + b2.
4x3 + 16x2 + 16x
4x(x2 + 4x + 4)
4x(x + 2)2
a = x, b = 2
Check 4x(x + 2)2 = 4x(x2 + 2x + 2x + 4)
= 4x(x2 + 4x + 4)
= 4x3 + 16x2 + 16x 

8. Check It Out! Example 2b
Factor each polynomial completely. Check your answer.
2x2y – 2y3
Factor out the GCF. 2y(x2 – y2) is a perfect-square trinomial of the form a2 – b2.
2x2y – 2y3
2y(x2 – y2)
2y(x + y)(x – y)
a = x, b = y
Check 2y(x + y)(x – y) = 2y(x2 + xy – xy – y2)
= 2x2y +2xy2 – 2xy2 – 2y3
= 2x2y – 2y3

9. 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

10. Example 3A: Factoring by Multiple Methods
Factor each polynomial completely.
9x2 + 3x – 2
The GCF is 1 and there is no pattern.
9x2 + 3x – 2
( x + )( x + )
a = 9 and c = –2;
Outer + Inner = 3 
Factors of 9 Factors of –2 Outer + Inner
1 and 9
1 and –2
1(–2) + 9(1) = 7
3 and 3
3(–2) + 3(1) = –3
–1 and 2
3(2) + 3(–1) = 3 
(3x – 1)(3x + 2)

11. Example 3B: Factoring by Multiple Methods
Factor each polynomial completely.
12b3 + 48b2 + 48b
The GCF is 12b; (b2 + 4b + 4) is a perfect-square trinomial in the form of
a2 + 2ab + b2.
12b(b2 + 4b + 4)
(b + )(b + )
Factors of 4 Sum
1 and
2 and  
a = 2 and c = 2
12b(b + 2)(b + 2)
12b(b + 2)2

12. Example 3C: Factoring by Multiple Methods
Factor each polynomial completely.
4y2 + 12y – 72
Factor out the GCF. There is no pattern. b = 3 and c = –18; look for factors of –18 whose sum is 3.
4(y2 + 3y – 18)
(y + )(y + ) 
Factors of –18 Sum
–1 and
–2 and 
–3 and
The factors needed are –3 and 6
4(y – 3)(y + 6)

13. Example 3D: Factoring by Multiple Methods.
Factor each polynomial completely.
(x4 – x2)
x2(x2 – 1)
Factor out the GCF.
x2(x + 1)(x – 1)
x2 – 1 is a difference of two squares.

14.   Check It Out! Example 3a Factor each polynomial completely.
3x2 + 7x + 4
a = 3 and c = 4;
Outer + Inner = 7
3x2 + 7x + 4
( x + )( x + ) 
Factors of 3 Factors of 4 Outer + Inner
3 and 1
1 and 4
3(4) + 1(1) = 13
2 and 2
3(2) + 1(2) = 8
4 and 1
3(1) + 1(4) = 7 
(3x + 4)(x + 1)

15.  Check It Out! Example 3b Factor each polynomial completely.
2p5 + 10p4 – 12p3
Factor out the GCF. There is no pattern. b = 5 and c = –6; look for factors of –6 whose sum is 5.
2p3(p2 + 5p – 6)
(p + )(p + ) 
Factors of – 6 Sum
– 1 and
The factors needed are –1 and 6
2p3(p + 6)(p – 1)

16.   Check It Out! Example 3c Factor each polynomial completely.
9q6 + 30q5 + 24q4
Factor out the GCF. There is no pattern.
3q4(3q2 + 10q + 8)
( q + )( q + )
a = 3 and c = 8;
Outer + Inner = 10 
Factors of 3 Factors of 8 Outer + Inner
3 and 1
1 and 8
3(8) + 1(1) = 25
2 and 4
3(4) + 1(2) = 14
4 and 2
3(2) + 1(4) = 10 
3q4(3q + 4)(q + 2)

17. Check It Out! Example 3d
Factor each polynomial completely.
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.

19. Lesson Quiz
Tell whether the polynomial is completely factored. If not, factor it.
1. (x + 3)(5x + 10) x2(x2 + 9)
no; 5(x+ 3)(x + 2)
completely factored
Factor each polynomial completely. Check your answer.
3. x3 + 4x2 + 3x x2 + 16x – 48
4(x + 6)(x – 2)
(x + 4)(x2 + 3)
5. 18x2 – 3x – 3
6. 18x2 – 50y2
3(3x + 1)(2x – 1)
2(3x + 5y)(3x – 5y)
7. 5x – 20x3 + 7 – 28x2
(1 + 2x)(1 – 2x)(5x + 7)