How do you divide (4x^4 -5x^2-2x+24)/((x + 4) )?

Oct 6, 2017

$4 {x}^{3} - 16 {x}^{2} + 59 x - 238 + \frac{976}{x + 4}$

Explanation:

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

$\text{consider the numerator}$

$\textcolor{red}{4 {x}^{3}} \left(x + 4\right) \textcolor{m a \ge n t a}{- 16 {x}^{3}} - 5 {x}^{2} - 2 x + 24$

$= \textcolor{red}{4 {x}^{3}} \left(x + 4\right) \textcolor{red}{- 16 {x}^{2}} \left(x + 4\right) \textcolor{m a \ge n t a}{+ 64 {x}^{2}} - 5 {x}^{2} - 2 x + 24$

$= \textcolor{red}{4 {x}^{3}} \left(x + 4\right) \textcolor{red}{- 16 {x}^{2}} \left(x + 4\right) \textcolor{red}{+ 59 x} \left(x + 4\right) \textcolor{m a \ge n t a}{- 236 x} - 2 x + 24$

=color(red)(4x^3)(x+4)color(red)(-16x^2)(x+4)color(red)(+59x)(x+4)color(red)
$\textcolor{w h i t e}{=} \textcolor{red}{- 238} \left(x + 4\right) \textcolor{m a \ge n t a}{+ 952} + 24$

$\text{quotient } = \textcolor{red}{4 {x}^{3} - 16 {x}^{2} + 59 x - 238}$

$\text{remainder } = 976$

$\Rightarrow \frac{4 {x}^{4} - 5 {x}^{2} - 2 x + 24}{x + 4}$

$= 4 {x}^{3} - 16 {x}^{2} + 59 x - 238 + \frac{976}{x + 4}$