# How do you divide ( 3x ^ { 3} + 11x ^ { 2} + 4x + 1) \div ( x ^ { 2} + x )?

Nov 3, 2017

The remainder is $= - 4 X + 1$ and the quotient is $= 3 X + 8$

#### Explanation:

$A = B Q + R$

$Q =$ is the quotient

$R$ is the remainder

Degree of $R$ is $<$ the degree of $B$

$A = 3 {X}^{3} + 11 {X}^{2} + 4 X + 1$

$B = {X}^{2} + X$

Perform a long division

$\textcolor{w h i t e}{a a a a}$$3 {X}^{3} + 11 {X}^{2} + 4 X + 1$$\textcolor{w h i t e}{a a a a}$$|$${X}^{2} + X$

$\textcolor{w h i t e}{a a a a}$$3 {X}^{3} + 3 {X}^{2}$$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$|$$3 X + 8$

$\textcolor{w h i t e}{a a a a}$$0 {X}^{3} + 8 {X}^{2} + 4 X$

$\textcolor{w h i t e}{a a a a a a a a a a}$$8 {X}^{2} + 8 X$

$\textcolor{w h i t e}{a a a a a a a a a a}$$0 {X}^{2} - 4 X + 1$

Therefore,

The remainder is $= - 4 X + 1$ and the quotient is $= 3 X + 8$

$\frac{3 {X}^{3} + 11 {X}^{2} + 4 X + 1}{{X}^{2} + X} = \left(3 X + 8\right) + \frac{- 4 X + 1}{{X}^{2} + X}$