# How do you find the quotient (3q^2+20q+11)div(q+6) using long division?

Apr 22, 2017

This is the same as long division. The only difference to the traditional method is the format.

$3 q + 2 - \frac{1}{q + 6}$

#### Explanation:

$\text{ } 3 {q}^{2} + 20 q + 11$
$\textcolor{m a \ge n t a}{3 q} \left(q + 6\right) \to \underline{3 {q}^{2} + 18 q} \leftarrow \text{ Subtract}$
$\text{ } \textcolor{w h i t e}{.} 0 + \textcolor{w h i t e}{2} 2 q + 11$
$\textcolor{m a \ge n t a}{\textcolor{w h i t e}{2} 2} \left(q + 6\right) \to \text{ "ul(2q+12 ) larr" Subtract}$
" "0color(magenta)(-1 larr" Remainder")

$\textcolor{m a \ge n t a}{3 q + 2 - \frac{1}{q + 6}}$

Apr 22, 2017

#### Explanation:

Given:

$\textcolor{w h i t e}{\frac{q + 6}{\textcolor{b l a c k}{q + 6}}} \frac{\textcolor{w h i t e}{\left(d + e + f\right)}}{\text{|} \textcolor{w h i t e}{x} 3 {q}^{2} + 20 q + 11}$

Find the first term in the quotient by dividing first term in the dividend by the first term in the divisor:

$\frac{3 {q}^{2}}{q} = 3 q$

Write the first term in the quotient:

$\textcolor{w h i t e}{\frac{q + 6}{\textcolor{b l a c k}{q + 6}}} \frac{3 q \textcolor{w h i t e}{d + e + f}}{\text{|} \textcolor{w h i t e}{x} 3 {q}^{2} + 20 q + 11}$

Multiply that term by the divisor $3 q \left(q + 6\right) = 3 {q}^{2} + 18 q$ subtract this from the dividend:

$\textcolor{w h i t e}{\frac{q + 6}{\textcolor{b l a c k}{q + 6}}} \frac{3 q \textcolor{w h i t e}{d + e + f}}{\text{|} \textcolor{w h i t e}{x} 3 {q}^{2} + 20 q + 11}$
color(white)("..........")ul(-3q^2-18q
$\textcolor{w h i t e}{\text{.........................}} 2 q + 11$

Find the next term in the quotient by dividing the first non-zero term in the results of the subtraction by the first term in the divisor:

$\frac{2 q}{q} = 2$

Write this term into the quotient:

$\textcolor{w h i t e}{\frac{q + 6}{\textcolor{b l a c k}{q + 6}}} \frac{3 q + 2 \textcolor{w h i t e}{e . + f}}{\text{|} \textcolor{w h i t e}{x} 3 {q}^{2} + 20 q + 11}$
color(white)("..........")ul(-3q^2-18q
$\textcolor{w h i t e}{\text{.........................}} 2 q + 11$

Multiply that term by the divisor $2 \left(q + 6\right) = 2 q + 12$ subtract this from the dividend:

$\textcolor{w h i t e}{\frac{q + 6}{\textcolor{b l a c k}{q + 6}}} \frac{3 q + 2 \textcolor{w h i t e}{e . + f}}{\text{|} \textcolor{w h i t e}{x} 3 {q}^{2} + 20 q + 11}$
color(white)("..........")ul(-3q^2-18q
$\textcolor{w h i t e}{\text{.........................}} 2 q + 11$
color(white)("......................")ul(-2q-12
$\textcolor{w h i t e}{\text{................................}} - 1$

Because the power of the divisor is greater than the power of the results of the subtraction, we know that we are done and that $- 1$ is a remainder.

The quotient can be expressed as:

$3 q + 2 - \frac{1}{q + 6}$