This document provides examples and instructions for factoring polynomials completely using different factoring methods such as greatest common factor, difference of squares, and grouping. It demonstrates how to determine if a polynomial is already fully factored or if additional factoring is possible. Students are shown step-by-step how to choose the appropriate factoring technique and combine methods as needed to fully factor polynomials that include multiple terms and variables. Practice problems with solutions are included to help students apply the factoring skills.

1. UNIT 8.7 FFAACCTTOORRIINNGG SSPPEECCIIAALL CCAASSEESS

2. Warm Up
Factor each trinomial.
1. x2 + 13x + 40
2. 5x2 – 18x – 8
(5x + 2)(x – 4)
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)
(3x + 5y)(3x – 5y)

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

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

5. 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.

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

7. 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.
5x2(x – 1) is completely factored.
B. (4x + 4)(x + 1)
4x + 4 can be further factored.
Factor out 4, the GCF of 4x and 4.
(4x + 4)(x + 1)
4(x + 1)(x + 1)
4(x + 1)2 is completely factored.

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

9. 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.
Factor remaining trinomial.
Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16)
= 10x2 + 40x + 8x + 32
= 10x2 + 48x + 32

10. 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

11. Check It Out! Example 2a
Factor each polynomial completely. Check
your answer.
4x3 + 16x2 + 16x
4x3 + 16x2 + 16x
4x(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 

12. 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

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

14. 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)

15. Example 3B: Factoring by Multiple Methods
Factor each polynomial completely.
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)
Factors of 4 Sum
1 and 4 5
2 and 2 4


12b(b + 2)(b + 2)
12b(b + 2)2

16. Example 3C: Factoring by Multiple Methods
Factor each polynomial completely.
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
(y + )(y + ) sum is 3.
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

17. 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.

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

19. Check It Out! Example 3b
Factor each polynomial completely.
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

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

21. 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.

23. Lesson Quiz
Tell whether the polynomial is completely
factored. If not, factor it.
1. (x + 3)(5x + 10) 2. 3x2(x2 + 9)
no; 5(x+ 3)(x + 2) completely factored
Factor each polynomial completely. Check
your answer.
3. x3 + 4x2 + 3x + 12 4. 4x2 + 16x – 48
(x + 4)(x2 + 3)
4(x + 6)(x – 2)
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)

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