# How do you use synthetic division to divide (12x^4 + 20x^3 - 24x^2 + 20x + 35)  by 3x+5?

Jul 11, 2018

The remainder is $- 65$ and the quotient is $= 4 {x}^{3} - 8 x + 20$

#### Explanation:

Let's perform the synthetic division

Divide by $3$

$\textcolor{w h i t e}{a a a a}$$- \frac{5}{3}$$|$$\textcolor{w h i t e}{a a a a}$$4$$\textcolor{w h i t e}{a a a a}$$\frac{20}{3}$$\textcolor{w h i t e}{a a a a a a}$$- 8$$\textcolor{w h i t e}{a a a a a a}$$\frac{20}{3}$$\textcolor{w h i t e}{a a a a}$$\frac{35}{3}$

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$- \frac{20}{3}$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a}$$\frac{40}{3}$$\textcolor{w h i t e}{a a a}$$- \frac{100}{3}$

$\textcolor{w h i t e}{a a a a a a a a a}$_________________________________________________________

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$4$$\textcolor{w h i t e}{a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a}$$- 8$$\textcolor{w h i t e}{a a a a a a a}$$20$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{- \frac{65}{3}}$

The remainder is $- 65$ and the quotient is $= 4 {x}^{3} + 0 {x}^{2} - 8 x + 20$

$\frac{12 {x}^{4} + 20 {x}^{3} - 24 {x}^{2} + 20 x + 35}{3 x + 5} = 4 {x}^{3} - 8 x + 20 - \frac{\frac{65}{3}}{x + \frac{5}{3}}$

$= 4 {x}^{3} - 8 x + 20 - \frac{65}{3 x + 5}$