# How do you divide ( -3x^3+ 16x^2-24x+9 )/(x + 1 )?

Nov 19, 2017

$- 3 {x}^{3} + 16 {x}^{2} - 24 x + 9$ is not divisible by $x + 1$

#### Explanation:

If we descompose $- 3 {x}^{3} + 16 {x}^{2} - 24 x + 9$ we obtain:
$\left(x - 3\right) \left(- 3 {x}^{2} + 7 x + 3\right)$ which is not divisible by $x + 1$.

Nov 19, 2017

$- 3 {x}^{2} + 19 x - 43 + \frac{52}{x + 1}$

#### Explanation:

$\text{one way is to use the divisor as a factor in the numerator}$

$\text{consider the numerator}$

$\textcolor{red}{- 3 {x}^{2}} \left(x + 1\right) \textcolor{m a \ge n t a}{+ 3 {x}^{2}} + 16 {x}^{2} - 24 x + 9$

$= \textcolor{red}{- 3 {x}^{2}} \left(x + 1\right) \textcolor{red}{+ 19 x} \left(x + 1\right) \textcolor{m a \ge n t a}{- 19 x} - 24 x + 9$

$= \textcolor{red}{- 3 {x}^{2}} \left(x + 1\right) \textcolor{red}{+ 19 x} \left(x + 1\right) \textcolor{red}{- 43} \left(x + 1\right) \textcolor{m a \ge n t a}{+ 43} + 9$

$= \textcolor{red}{- 3 {x}^{2}} \left(x + 1\right) \textcolor{red}{+ 19 x} \left(x + 1\right) \textcolor{red}{- 43} \left(x + 1\right) + 52$

$\text{quotient "=color(red)(-3x^2+19x-43)," remainder } = 52$

$\Rightarrow \frac{- 3 {x}^{3} + 16 {x}^{2} - 24 x + 9}{x + 1}$

$= - 3 {x}^{2} + 19 x - 43 + \frac{52}{x + 1}$