How do you use synthetic division to divide (3x^3-4x^2+5)/(x-3/2)?

Feb 21, 2017

The remainder is $= \frac{49}{8}$ and the quotient is $= 3 {x}^{2} + \frac{1}{2} x + \frac{3}{4}$

Explanation:

We perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$\frac{3}{2}$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$3$$\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}$$5$

$\textcolor{w h i t e}{a a a a a a}$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a}$$\textcolor{w h i t e}{a a a a a a a a}$$\frac{9}{2}$$\textcolor{w h i t e}{a a a a}$$\frac{3}{4}$$\textcolor{w h i t e}{a a a a}$$\frac{9}{8}$

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

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{w h i t e}{a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a}$$\frac{1}{2}$$\textcolor{w h i t e}{a a a a a}$$\frac{3}{4}$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{\frac{49}{8}}$

The remainder is $= \frac{49}{8}$

The quotient is $= 3 {x}^{2} + \frac{1}{2} x + \frac{3}{4}$

Verification, by using the remainder theorem

$f \left(\frac{3}{2}\right) = \frac{81}{8} - 9 + 5 = \frac{81}{8} - 4 = \frac{49}{8}$