Factoring

Difference of Cubes

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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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