# Given polynomial f(x)=x^3+2x^2-51x+108 and a factor x+9 how do you find all other factors?

Dec 31, 2016

The answer is $= \left(x + 9\right) \left(x - 4\right) \left(x - 3\right)$

#### Explanation:

$f \left(x\right) = {x}^{3} + 2 {x}^{2} - 51 x + 108$

$\left(x + 9\right)$ is a factor

We do a long division

$\textcolor{w h i t e}{a a a a}$${x}^{3} + 2 {x}^{2} - 51 x + 108$$\textcolor{w h i t e}{a a a a}$∣$x + 9$

$\textcolor{w h i t e}{a a a a}$${x}^{3} + 9 {x}^{2}$$\textcolor{w h i t e}{a a a a a a a a a a a a a a a}$∣${x}^{2} - 7 x + 12$

$\textcolor{w h i t e}{a a a a}$$0 - 7 {x}^{2} - 51 x$

$\textcolor{w h i t e}{a a a a a a}$$- 7 {x}^{2} - 63 x$

$\textcolor{w h i t e}{a a a a a a a a}$$- 0 + 12 x + 108$

$\textcolor{w h i t e}{a a a a a a a a a a a a}$$+ 12 x + 108$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$+ 0 + 0$

Therefore,

$\frac{{x}^{3} + 2 {x}^{2} - 51 x + 108}{x + 9} = {x}^{2} - 7 x + 12$

We can factorise the quotient

${x}^{2} - 7 x + 12 = \left(x - 4\right) \left(x - 3\right)$