# How do you list all possible roots and find all factors and zeroes of x^4-x^3+14x^2-16x-32?

Jul 5, 2018

#### Explanation:

The function is

$P \left(x\right) = {x}^{4} - {x}^{3} + 14 {x}^{2} - 16 x - 32$

$= {x}^{4} - {x}^{3} - \textcolor{red}{2 {x}^{2}} + 14 {x}^{2} + \textcolor{red}{2 {x}^{2}} - 16 x - 32$

$= \left({x}^{4} - {x}^{3} - 2 {x}^{2}\right) + \left(16 {x}^{2} - 16 x - 32\right)$

$= {x}^{2} \left({x}^{2} - x - 2\right) + 16 \left({x}^{2} - x - 2\right)$

$= \left({x}^{2} + 16\right) \left({x}^{2} - x - 2\right)$

$= \left({x}^{2} + 16\right) \left(x + 1\right) \left(x - 2\right)$

The real roots are $\left(x = - 1\right)$ and $\left(x = 2\right)$ and the imaginary roots are $\left(x = 4 i\right)$ and $\left(x = - 4 i\right)$

graph{x^4-x^3+14x^2-16x-32 [-36.53, 36.53, -18.27, 18.28]}