Question

How do you factor `x^3+343`?

Answer 1

`x^3+343=(x+7)(x^2-7x+49)`

Explanation:

The sum of cubes identity can be written:

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

We can use this with `a=x` and `b=7` as follows:

`x^3+343`

`=x^3+7^3`

`=(x+7)(x^2-7x+7^2)`

`=(x+7)(x^2-7x+49)`

The remaining quadratic factor can only be factored further with Complex coefficients, which we can express in terms of the primitive Complex cube root of `1`:

`=(x+7)(x+7omega)(x+7omega^2)`

where `omega = -1/2+sqrt(3)/2i`