# How do you find the quotient of (6x^3+16x^2-60x+39)div(2x+10)?

Apr 30, 2017

The quotient is $= \left(3 {x}^{2} - 7 x + 5\right)$

#### Explanation:

We can perform either a long division or a synthetic division

Let $f \left(x\right) = 6 {x}^{3} + 16 {x}^{2} - 60 x + 39$

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$- 5$$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$6$$\textcolor{w h i t e}{a a a a a a}$$16$$\textcolor{w h i t e}{a a a a a}$$- 60$$\textcolor{w h i t e}{a a a a a}$$39$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$_________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a}$$- 30$$\textcolor{w h i t e}{a a a a a a}$$70$$\textcolor{w h i t e}{a a a a a}$$- 50$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$6$$\textcolor{w h i t e}{a a a a a a}$$- 14$$\textcolor{w h i t e}{a a a a a a}$$10$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{- 11}$

$f \left(x\right) = \left(6 {x}^{2} - 14 x + 10\right) - \frac{11}{x + 5}$

or

$f \left(x\right) = 3 {x}^{2} - 7 x + 5 - \frac{11}{2 x + 10}$

The remainder is $= - 11$

Apr 30, 2017

color(blue)(3x^2-7x+5 and a remainder of color(blue)(-11

#### Explanation:

 color(white)(...............................)color(blue)(3x^2-7x+5
$\textcolor{w h i t e}{a} 2 x + 10$$|$ color(white)(.)overline(6x^3+16x^2-60x+39
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots .} \underline{6 {x}^{3} + 30 {x}^{2}}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} - 14 {x}^{2} - 60 x$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{- 14 {x}^{2} - 70 x}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . .} 10 x + 39$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{10 x + 50}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 0 - 11$