# How do you divide (-4x^3-15x^2-4x-12)/(x-4) ?

Jul 20, 2018

The remainder is $= - 524$ and the quotient is $= - 4 {x}^{2} - 31 x - 128$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$4$$|$$\textcolor{w h i t e}{a a a a}$$- 4$$\textcolor{w h i t e}{a a a a}$$- 15$$\textcolor{w h i t e}{a a a a a a}$$- 4$$\textcolor{w h i t e}{a a a a a}$$- 12$

$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$- 16$$\textcolor{w h i t e}{a a a a}$$- 124$$\textcolor{w h i t e}{a a a a}$$- 512$

$\textcolor{w h i t e}{a a a a a a a a a}$_________

$\textcolor{w h i t e}{a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$- 4$$\textcolor{w h i t e}{a a a a}$$- 31$$\textcolor{w h i t e}{a a a a}$$- 128$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{- 524}$

The remainder is $= - 524$ and the quotient is $= - 4 {x}^{2} - 31 x - 128$

ALSO,

Apply the remainder theorem

When a polynomial $f \left(x\right)$ is divided by $\left(x - c\right)$, we get

$f \left(x\right) = \left(x - c\right) q \left(x\right) + r$

Let $x = c$

Then,

$f \left(c\right) = 0 + r$

Here,

$f \left(x\right) = - 4 {x}^{3} - 15 {x}^{2} - 4 x - 12$

Therefore,

$f \left(4\right) = - 4 \cdot {4}^{3} - 15 \cdot {4}^{2} - 4 \cdot 4 - 12$

$= - 256 - 240 - 16 - 12$

$= - 524$

The remainder is $= - 524$