Factoring

Sum 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 Sum 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 Sum of cubes, we factor it into a binomial and a trinomial. The binomial being the cube root of the first term plus 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 subtraction, 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. 8x3 + 27: solution

2. 54x7 + 16x: solution

3. 3x5 - 3x2: solution

#1 solution

1. 8x3 + 27

We first check that we have a sum of cubes.
since 8x3 and 27 are both perfect cubes we do have a sum of cubes.
the cube root of 8x3 is 2x, and the cube root of 27 is 3
so our binomial is (2x + 3)
we now use the binomial to create the trinomial
we square the first term 2x to get 4x2 as our first term
we change the sign from + to - and multiply 2x and 3 to get -6x as our middle term
we square the second term 3 to get 9 as our third term
so the trinomial is (4x2 - 6x + 9)
thus our answer is (2x + 3)(4x2 - 6x + 9)

#2 solution

2. 54x7 + 16x

In checking if we have a sum of cubes, we see neither 54x7 or 16x are perfect cubes
But we have a common factor we must factor out first, 2x
this gives us 2x(27x6 + 8)
Now both 27x6 and 8 are perfect cubes
the cube root of 27x6  is 3x2 , and the cube root of 8 is 2
so our binomial is (3x2 + 2)
Now to get the trinomial we square 3x2 to get 9x4 as our first term
we change the sign to - and multiply 3x2 and 2 to get -6x2 as the middle term
we square 2 to get 4 as our third term
so the trinomial is (9x4 - 6x2 + 4)
and the answer is 2x(3x2 + 2)(9x4 - 6x2 + 4)

#3 solution

3. 3x5 + 3x2

Here again we do not have a sum of cubes, but we do have a common factor 3x2
3x2 (x3 + 1), now both x3 and 1 are perfect cubes
the cube root of x3 is x and the cube root of 1 is 1
so our binomial is (x + 1)
to get the first term of our trinomial we square x: x2
to get the second term we change the sign to - and multiply x and 1: - x
to get the third term we square 1 : 1
so our trinomial is (x2 -x + 1)
and the answer is 3x2 (x + 1)( x2 - x + 1)