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

Jul 9, 2018

The quotient is $= {d}^{3} - 4 {d}^{2} - 8 d - 15$

#### Explanation:

Perform a long division to obtain the quotient

$\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 a a}$$|$$d - 2$

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

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

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

The quotient is $= {d}^{3} - 4 {d}^{2} - 8 d - 15$

The remainder is $= - 13$

$\frac{{d}^{4} - 6 {d}^{3} + 0 {d}^{2} + d + 17}{d - 2} = {d}^{3} - 4 {d}^{2} - 8 d - 15 - \frac{13}{d - 2}$