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Sum of Cubes

Reference > Mathematics > Algebra > Factoring Higher Degree Polynomials
 

In the previous section, we learned how to factor a binomial in which the two terms are perfect cubes, and they are being subtracted. In this section, we'll learn how to factor a similar binomial, in which the two perfect cubes are being added. First, we'll introduce this topic by asking you to perform another polynomial multiplication:

(a + b)(a2 - ab + b2)
a(a2 - ab + b2) +b(a2 - ab + b2)
a3 - a2b +ab2 +a2b - ab2 + b3
a3 + b3

And thus we have our factoring rule:

Sum of Cubes
a3 + b3 = (a + b)(a2 - ab + b2)

If you did the section on difference of cubes, you shouldn't need too many examples of how to use this rule, since it is almost identical to the Difference of Cubes rule. Here's one example:

Example
Factor 128x6 + 2y9

Solution
First we note that 2 can be factored out of the entire expression:

128x6 + 2y9 = 2(64x6 + y9)

Since 64x6 and y9 are both perfect cubes, a = 4x2 and b = y3

128x6 + 2y9 = 2(4x2 + y3)((4x2)2 - 4x2(y3) + (y3)2)
128x6 + 2y9 = 2(4x2 + y3)(16x4 - 4x2y3 + y6)

As with the Difference of Cubes rule, we can do a cursory check of our answer by multiplying the first terms of the two polynomials, and multiplying the last terms of the two polynomials:

4x2(16x4) = 64x6
y3(y6) = y9

These match the two terms in the original binomial, which suggests that we're on the right track. It's still possible we have errors, and if you are uncertain, you can do a full check by completely multiplying out your polynomials to see if you get back to the original expression.

Questions

1.
Factor x3 + 8y3
2.
Factor 270x4 + 10x
3.
Factor x3r + y6rz12r
4.
Factor 729x3y3 + z3
5.
Factor 1,000,001 (hint - write it as 1003 + 13)
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Difference of CubesDifference of Cubes
Sum or Difference of Odd PowersSum or Difference of Odd Powers
 

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