**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

A |

1. x^{3} - 27: solution

2. 8x^{6} - 125: solution

3. 250x^{4} - 128x: solution

1. x^{3} - 27

We first check that we have a difference of cubes

since x^{3} 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 x^{2}

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 (x^{2} + 3x + 9)

and the answer is (x-3)(x^{2} + 3x + 9)

2. 8x^{6} - 125

We first check that we have a sum of cubes.

since 8x^{6} and 125 are both perfect cubes we do have a sum of cubes.

the cube root of 8x^{6} is 2x^{2}, and the cube root of 15 is 5

so our binomial is (2x^{2} - 5)

we now use the binomial to create the trinomial

we square the first term 2x^{2} to get 4x^{4} as our first term

we change the sign from - to + and multiply 2x^{2} and 5 to get +10x^{2} as our middle term

we square the second term 5 to get 25 as our third term

so the trinomial is (4x^{4} + 10x^{2} + 25)

thus our answer is (2x - 5)(4x^{2} + 10x^{2} + 25)

3. 250x^{4} - 128x

In checking if we have a sum of cubes, we see neither 250x^{4} or 128x are perfect cubes

But we have a common factor we must factor out first, 2x

this gives us 2x(125x^{3} - 64)

Now both 125x^{3} and 64 are perfect cubes

the cube root of 125x^{3} 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 25x^{2} 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 (25x^{2} + 20x + 16)

and the answer is 2x(5x - 4)(25x^{2} + 20x + 16)