diffcube

Factoring

Difference of Cubes

A binomial is factorable only if it is one of three things a Difference of Squares, a Difference of Cubes, or a Sum of Cubes. A binomial is a Difference of Cubes if both terms are perfect cubes. Recall we may have to factor out a common factor first.

If we determine that a binomial is a difference of cubes, we factor it into a binomial and a trinomial. The binomial being the cube root of the first term minus the cube root of the second term. The trinomial comes from the binomial. We square the first term of the binomial, change the sign to addition, multiply the two terms together, and square the second term of the binomial, as in the following formula

A³ - B³ = (A - B)(A² + AB + B²)

Factor each of the following.

1. x³ - 27: solution

2. 8x⁶ - 125: solution

3. 250x⁴ - 128x: solution

#1 solution

1. x³ - 27

We first check that we have a difference of cubes
since x³ and 27 are perfect cubes, we do
the cube root of x is x and the cube root of 27 is 3
so our binomial is (x-3)
to get the first term of the trinomial we square x getting x²
to get the second term of the trinomial we change the sign to + and multiply x by 3, getting +3x
to get the third term of the trinomial we square 3 getting 9
so our trinomial is (x² + 3x + 9)
and the answer is (x-3)(x² + 3x + 9)

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#2 solution

2. 8x⁶ - 125

We first check that we have a sum of cubes.
since 8x⁶ and 125 are both perfect cubes we do have a sum of cubes.
the cube root of 8x⁶ is 2x², and the cube root of 15 is 5
so our binomial is (2x² - 5)
we now use the binomial to create the trinomial
we square the first term 2x² to get 4x⁴ as our first term
we change the sign from - to + and multiply 2x² and 5 to get +10x² as our middle term
we square the second term 5 to get 25 as our third term
so the trinomial is (4x⁴ + 10x² + 25)
thus our answer is (2x - 5)(4x² + 10x² + 25)

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#3 solution

3. 250x⁴ - 128x

In checking if we have a sum of cubes, we see neither 250x⁴ or 128x are perfect cubes
But we have a common factor we must factor out first, 2x
this gives us 2x(125x³ - 64)
Now both 125x³ and 64 are perfect cubes
the cube root of 125x³ is 5x , and the cube root of 64 is 4
so our binomial is (5x - 4)
Now to get the trinomial we square 5x to get 25x² as our first term
we change the sign to + and multiply 5x and 4 to get +20x as the middle term
we square 4 to get 16 as our third term
so the trinomial is (25x² + 20x + 16)
and the answer is 2x(5x - 4)(25x² + 20x + 16)

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