Question

How do you factor the expression `x^3+1/8`?

Answer 1

`x^3 + 1/8 = (x + 1/2)(x^2 -1/2x + 1/4)`

Explanation:

The sum of cubes formula states that

`a^3 + b^3 = (a+b)(a^2-ab+b^2)`

(Verify this by multiplying it out)

As `1/8 = (1/2)^3` we can apply the formula.

`x^3 + 1/8 = x^3 + (1/2)^3 = (x + 1/2)(x^2 -1/2x + 1/4)`

Note that we cannot factor any further, as the discriminant for the remaining quadratic is negative:
`(-1/2)^2 - 4(1)(1/4) = 1/4 - 1 = -3/4`
and thus it has no real roots.