Question

How do you factor x^3+27?

Answer 1

Because factoring x^3+27 is the same as finding where the graph passes through the x axis, we can just set the equation equal to zero and solve. `f(x) = (x+3)(x^2-3x+9)`

Explanation:

Let f(x) = x³ + 27
0 = x³ + 27
x³= -27
x = -3
This means that x = -3 is the only zero of the graph of f(x). Since we know `(x+3)` is one factor of `f(x)`, to find the 2nd factor,
`(x^3+27)/(x+3)`
We get the second factor to be `x^2-3x+9`.

Therefore, `f(x) = (x+3)(x^2-3x+9)`

Answer 2

`x^3+27 = (x+3)(x^2-3x+9)`

Explanation:

The sum of cubes identity can be written:

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

Note that both `x^3` and `27=3^3` are perfect cubes, so we can use the sum of cubes identity with `a=x` and `b=3` as follows:

`x^3+27 = x^3+3^3`

`color(white)(x^3+27) = (x+3)(x^2-3x+3^2)`

`color(white)(x^3+27) = (x+3)(x^2-3x+9)`

The remaining quadratic has no simpler linear factors with Real coefficients.