# What is the quotient of d-2 divided by d^4-6d^3+d+17?

Jul 3, 2017

The quotient is $= \left({d}^{3} - 4 {d}^{2} - 8 d - 15\right)$

#### Explanation:

Let's perform the long division

$d - 2$$\textcolor{w h i t e}{a a a a}$$|$${d}^{4} - 6 {d}^{3} + 0 {d}^{2} + d + 17$$\textcolor{w h i t e}{a a}$$|$${d}^{3} - 4 {d}^{2} - 8 d - 15$

$\textcolor{w h i t e}{a a a a a a a a a a}$${d}^{4} - 2 {d}^{3}$

$\textcolor{w h i t e}{a a a a a a a a a a a}$$0 - 4 {d}^{3} + 0 {d}^{2}$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- 4 {d}^{3} + 8 {d}^{2}$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$- 0 - 8 {d}^{2} + d$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a}$$- 8 {d}^{2} + 16 d$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a}$$- 0 - 15 d + 17$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a}$$- 15 d + 30$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a}$$- 0 - 13$

Therefore,

$\frac{{d}^{4} - 6 {d}^{3} + d + 17}{d - 2} = {d}^{3} - 4 {d}^{2} - 8 d - 15 - \frac{13}{d - 2}$

The remainder is $= - 13$ and the quotient is $= \left({d}^{3} - 4 {d}^{2} - 8 d - 15\right)$