What is the method to factor $x^3 + 1$?

Question

In the solution to a problem, it's stated that

We see that $x^3+1=(x+1)(x^2-x+1)$.

Why is this, and what method can I use for similar problems with different coefficients?

The full problem is

Find the remainder when $x^{81}+x^{48}+2x^{27}+x^6+3$ is divided by $x^3+1$.

Answer 1

For the full problem: let be $$y=x^3.$$ Your problem is equivalent to find the remainder of $$ P(y) = y^{27} + y^{16} + 2y^9 + y^2 + 3 $$ when it is divided by $$y+1.$$ By Remainder's Theorem, that remainder is given by $$P(-1)=2.$$

Answer 2

$x^3+1=(x+1)(x^2-x+1)$ is something one just knows, similar to how one knows that $ x^2+2x+1=(x+1)^2 $, or $ x^2-1=(x+1)(x-1) $

If you don't already know it, note that $(-1)^3+1=0$, which implies that $x-(-1)$ is a factor of $x^3+1$. Finding the second factor can be done through long division.

Answer 3

For the full problem, working modulo $x^3+1$ we have $$x^3=-1\implies x^{3n}=(-1)^n\implies x^{81}+x^{48}+2x^{27}+x^6+3=-1+1-2+1+3=2.$$

So no such factorisation is needed. But when it is needed, here's how to do it. By the factor theorem, the fact that $(-1)^3+1=0$ implies $x-(-1)=x+1$ is a factor. So try $x^3+1=(x+1)(ax^2+bx+c)$. We know $a=1$ from the $x^3$ coefficient and $c=1$ from the constant term, and the $x^2$ coefficient tells us $0=a+b\implies b=-1$.

Answer 4

$$x^3+1=(x+1)(x^2-x+1)$$ can be seen from our good friend geometric series: $$\dfrac{r^n-1}{r-1} = 1+r+r^2+\cdots+r^{n-1}$$

Plug $r=-x$ and $n=3$

Answer 5

Since $(-1)^3+1=0,$ we know that $x^3+1$ is divisible by $x+1$, which gives the following way: $$x^3+1=x^3+x^2-x^2-x+x+1=x^2(x+1)-x(x+1)+x+1=(x+1)(x^2-x+1).$$