Complete Factorization of
Polynomials
Brenda Meery
Kaitlyn Spong
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Printed: April 16, 2015
AUTHORS
Brenda Meery
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Complete Factorization of Polynomials
1
Complete Factorization of
Polynomials
Here you will learn how to factor a polynomial completely by first looking for common factors and then factoring
the resulting expression.
Can you factor the following polynomial completely?
8x3 + 24x2 − 32x
Watch This
Khan Academy Factoring and the Distributive Property
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/59347
Guidance
A cubic polynomial is a polynomial of degree equal to 3. Examples of cubics are:
• 9x3 + 10x − 5
• 8x3 + 2x2 − 5x − 7
Recall that to factor a polynomial means to rewrite the polynomial as a product of other polynomials . You will not
be able to factor all cubics at this point, but you will be able to factor some using your knowledge of common factors
and factoring quadratics. In order to attempt to factor a cubic, you should:
1. Check to see if the cubic has any common factors. If it does, factor them out.
2. Check to see if the resulting expression can be factored, especially if the resulting expression is a quadratic.
To factor the quadratic expression you could use the box method, or any method you prefer.
Anytime you are asked to factor completely, you should make sure that none of the pieces (factors) of your final
answer can be factored any further. If you follow the steps above of first checking for common factors and then
checking to see if the resulting expressions can be factored, you can be confident that you have factored completely.
Example A
Factor the following polynomial completely: 3x3 − 15x.
Solution: Look for the common factors in each of the terms. The common factor is 3x. Therefore:
3x3 − 15x = 3x(x2 − 5)
The resulting quadratic, x2 − 5, cannot be factored any further (it is NOT a difference of perfect squares). Your
answer is 3x(x2 − 5).
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Example B
Factor the following polynomial completely: 2a3 + 16a2 + 30a.
Solution: Look for the common factors in each of the terms. The common factor is 2a. Therefore:
2a3 + 16a2 + 30a = 2a(a2 + 8a + 15)
The resulting quadratic, a2 +8a+15 can be factored further into (a+5)(a+3). Your final answer is 2a(a+5)(a+3).
Example C
Factor the following polynomial completely: 6s3 + 36s2 − 18s − 42.
Solution: Look for the common factors in each of the terms. The common factor is 6. Therefore:
6s3 + 36s2 − 18s − 42 = 6(s3 + 6s2 − 3s − 7)
The resulting expression is a cubic, and you don’t know techniques for factoring cubics without common factors at
this point. Therefore, your final answer is 6(s3 + 6s2 − 3s − 7).
Note: It turns out that the resulting cubic cannot be factored, even with more advanced techniques. Remember that
not all expressions can be factored. In fact, in general most expressions cannot be factored.
Concept Problem Revisited
Factor the following polynomial completely: 8x3 + 24x2 − 32x.
Look for the common factors in each of the terms. The common factor is 8x. Therefore:
8x3 + 24x2 + 32x = 8x(x2 + 3x − 4)
The resulting quadratic can be factored further into (x + 4)(x − 1). Your final answer is 8x(x + 4)(x − 1).
Guided Practice
Factor each of the following polynomials completely.
1. 9w3 + 12w.
2. y3 + 4y2 + 4y.
3. 2t 3 − 10t 2 + 8t.
Answers:
1. The common factor is 3w. Therefore, 9w3 + 12w = 3w(3w2 + 4). The resulting quadratic cannot be factored any
further, so your answer is 3w(3w2 + 4).
2. The common factor is y. Therefore, y3 + 4y2 + 4y = y(y2 + 4y + 4). The resulting quadratic can be factored into
(y + 2)(y + 2) or (y + 2)2 . Your answer is y(y + 2)2 .
3. The common factor is 2t. Therefore, 2t 3 − 10t 2 + 8t = 2t(t 2 − 5t + 4). The resulting quadratic can be factored
into (t − 4)(t − 1). Your answer is 2t(t − 4)(t − 1).
Explore More
Factor each of the following polynomials completely.
1. 6x3 − 12
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2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Chapter 1. Complete Factorization of Polynomials
4x3 − 12x2
8y3 + 32y
15a3 + 30a2
21q3 + 63q
4x3 − 12x2 − 8
12e3 + 6e2 − 6e
15s3 − 30s + 45
22r3 + 66r2 + 44r
32d 3 − 16d 2 + 12d
5x3 + 15x2 + 25x − 30
3y3 − 18y2 + 27y
12s3 − 24s2 + 36s − 48
8x3 + 24x2 − 80x
5x3 − 25x2 − 70x
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