# What is the answer to 4s5+8s4+5s3+10s2? Factor it

Apr 9, 2018

#### Explanation:

$f \left(s\right) = 4 {s}^{5} + 8 {s}^{4} + 5 {s}^{3} + 10 {s}^{2}$
$f \left(s\right) = {s}^{2} \left(4 {s}^{3} + 8 {s}^{2} + 5 s + 10\right)$

After factoring out ${s}^{2}$ we are left with a polynomial of degree $3$ to factorise $g \left(s\right) = 4 {s}^{3} + 8 {s}^{2} + 5 s + 10$. This can be done using the factor theorem.

After testing some integers it can be found that:
$g \left(- 2\right) = 0$

Hence $\left(s + 2\right)$ is a factor of $g \left(s\right)$ and can be factored out by long division. This gives the result:
$g \left(s\right) = \left(s + 2\right) \left(4 {s}^{2} + 5\right)$

$4 {s}^{2} + 5$ can be factorised further using the quadratic formula.
$s = \frac{- 0 \pm \sqrt{{0}^{2} - 4 \times 4 \times 5}}{2 \times 4}$
$s = \pm \frac{\sqrt{- 80}}{8}$
$s = \pm i \frac{\sqrt{5}}{2}$

Hence
$g \left(s\right) = \left(s + 2\right) \left(s + i \frac{\sqrt{5}}{2}\right) \left(s - i \frac{\sqrt{5}}{2}\right)$

$4 {s}^{5} + 8 {s}^{4} + 5 {s}^{3} + 10 {s}^{2} = {s}^{2} \left(s + 2\right) \left(s + i \frac{\sqrt{5}}{2}\right) \left(s - i \frac{\sqrt{5}}{2}\right)$