# How do you use the remainder theorem and synthetic division to find the remainder when 2x^3-7x^2 div x-5?

Feb 20, 2017

The remainder is $= 75$

#### Explanation:

The remainder theorem states that when a polynomial $f \left(x\right)$ is divided by $x - c$

$f \left(x\right) = \left(x - c\right) q \left(x\right) + r \left(x\right)$

$f \left(c\right) = 0 + r$

Here,

$f \left(x\right) = 2 {x}^{3} - 7 {x}^{2}$

and $\left(x - 5\right)$

$f \left(5\right) = 2 \cdot 125 - 175 = 250 - 175 = 75$

The remainder is $= 75$

We now 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}$$|$$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$- 7$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$0$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a a a}$$\textcolor{w h i t e}{a a a a}$$10$$\textcolor{w h i t e}{a a a a}$$15$$\textcolor{w h i t e}{a a a a}$$75$

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

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$\textcolor{w h i t e}{a a a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$15$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{75}$

The remainder is also $= 75$

The quotient is $= 2 {x}^{2} + 3 x + 15$