How do you factor x^3 - 16?

Apr 12, 2015

Set this expression to $0$ to determine a root; then use synthetic division to determine the second factor

If ${x}^{3} - 16 = 0$
then
${x}^{3} = 16$
and
$x = 2 \sqrt[3]{2}$

So
$\left(x - 2 \sqrt[3]{2}\right)$
is a factor of ${x}^{3} - 16$

Use synthetic division to divide $\left(x - 2 \sqrt[3]{3}\right)$
into $\left({x}^{3} - 16\right)$
giving
$\left({x}^{2} + 2 \sqrt[3]{2} x + 4 \sqrt[3]{4}\right)$

So
$\left({x}^{3} - 16\right)$
=(x-2root(3)(2))(x^2+2root(3)(2)x + 4root(3)4