Factorization of Special Cubics
Brenda Meery
Kaitlyn Spong
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Printed: April 23, 2015
AUTHORS
Brenda Meery
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Factorization of Special Cubics
1
Factorization of Special
Cubics
Here you’ll learn to factor the sum and difference of perfect cubes.
Factor the following cubic polynomial: 375x3 + 648.
Watch This
James Sousa: Factoring Sum and Difference of Cubes
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/63481
Guidance
While many cubics cannot easily be factored, there are two special cases that can be factored quickly. These special
cases are the sum of perfect cubes and the difference of perfect cubes.
• Factoring the sum of two cubes follows this pattern:
x3 + y3 = (x + y)(x2 − xy + y2 )
• Factoring the difference of two cubes follows this pattern:
x3 − y3 = (x − y)(x2 + xy + y2 )
The acronym SOAP can be used to help you remember the positive and negative signs when factoring the sum and
difference of cubes. SOAP stands for " Same", " Opposite", " Always Positive". "Same" refers to the first sign in
the factored form of the cubic being the same as the sign in the original cubic. "Opposite" refers to the second sign
in the factored cubic being the opposite of the sign in the original cubic. "Always Positive" refers to the last sign in
the factored form of the cubic being always positive. See below:
1
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Example A
Factor: x3 + 27.
Solution: This is the sum of two cubes and uses the factoring pattern: x3 + y3 = (x + y)(x2 − xy + y2 ).
x3 + 33 = (x + 3)(x2 − 3x + 9).
Example B
Factor: x3 − 343.
Solution: This is the difference of two cubes and uses the factoring pattern: x3 − y3 = (x − y)(x2 + xy + y2 ).
x3 − 73 = (x − 7)(x2 + 7x + 49).
Example C
Factor: 64x3 − 1.
Solution: This is the difference of two cubes and uses the factoring pattern: x3 − y3 = (x − y)(x2 + xy + y2 ).
(4x)3 − 13 = (4x − 1)(16x2 + 4x + 1).
Concept Problem Revisited
Factor the following cubic polynomial: 375x3 + 648.
First you need to recognize that there is a common factor of 3. 375x3 + 648 = 3(125x3 + 216)
Notice that the result is the sum of two cubes. Therefore, the factoring pattern is x3 + y3 = (x + y)(x2 − xy + y2 ).
375x3 + 648 = 3(5x + 6)(25x2 − 30x + 36)
Guided Practice
Factor each of the following cubics.
1. x3 + 512
2. 8x3 + 125
3. x3 − 216
Answers:
1. x3 + 83 = (x + 8)(x2 − 8x + 64).
2. (2x)3 + 53 = (2x + 5)(4x2 − 10x + 25).
3. x3 − 63 = (x − 6)(x2 + 6x + 36).
Explore More
Factor each of the following cubics.
1. x3 + h3
2. a3 + 125
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Chapter 1. Factorization of Special Cubics
8x3 + 64
x3 + 1728
2x3 + 6750
h3 − 64
s3 − 216
p3 − 512
4e3 − 32
2w3 − 250
x3 + 8
y3 − 1
125e3 − 8
64a3 + 2197
54z3 + 3456
3