Question

How do you factor $x^3+1^3$?

Answer 1

Finding the factor of $x^3+1^3$:

Use the identity $a^3+b^3=(a+b)(a^2-ab+b^2)$ to factorize $x^3+1^3$.

Compare $x^3+1^3$ with $(a^3+b^3)$.

we get, $a=x$ and $b=1$.

Therefore,

$x^3+1^3$ $=$ $(x+1)(x^2-x+1)$

$x^2-x+1$ is a quadratic equation and its discriminant $D=(b^2-4ac)$ is $-3$, which is negative.

$\because D<0$, it means $x^2-x+1$ has no real roots.

So $x^2-x+1$ can not be factorized further.

$\Rightarrow x^3+1^3$ $=$ $(x+1)(x^2-x+1)$

Therefore, the factors of $x^3+1^3$ is $(x+1)(x^2-x+1)$.