Question

How do you factor completely `64 + a^3`?

Answer 1

`a^3+64=(a+4)(a^2-4a+16)`

Explanation:

The expression `a^3+64=a^3+4^3` is the sum of two cubes (3rd powers). In general, if you have an arbitrary sum of two cubes, say `x^3+y^3`, it can be factored as:

`x^3+y^3=(x+y)(x^2-xy+y^2)`. You should check this by expansion (multiplication) of the right-hand side.

Now apply this formula to the problem at hand by substituting `x=a` and `y=4`.

The difference of two cubes can also be factored, and the formula for the sum of two cubes can be used to do it:

`x^3-y^3=x^3+(-y)^3`

`=(x+(-y))(x^2-x(-y)+(-y)^2)`

`=(x-y)(x^2+xy+y^2)`.