# Math Help - How to factor (x^3 + 8) ?

1. ## How to factor (x^3 + 8) ?

How do I factor (x^3 + 8) ?

Any help would be greatly appreciated!

2. Originally Posted by s3a
How do I factor (x^3 + 8) ?

Any help would be greatly appreciated!
Notice that

$x^3 + 8 = x^3 + 2^3$, a sum of two cubes.

So it can be factorised using the sum of two cubes rule...

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Therefore

$x^3 + 8 = (x + 2)(x^2 - 2x + 4)$.

3. Let $f(x)=x^{3}+8$

Since $f(-2)=0$, by factor theorem, $(x+2)$ is a factor.

So $x^{3}+8=(x+2)(ax^{2}+bx+c)$

Comparing coeff. of $x^{3}$, $a=1$

Comparing constants, $c=4$

Comparing coeff. of $x^{2}$, $b+2a=0$
$b=-2$

Note: you can find the quadratic factor by long division also.

Thus
$f(x)=(x+2)(x^{2}-2x+4)$