How do you evaluate \frac{4x^{4}-5x^{2}+2x-10}{2x-3}?

Aug 11, 2018

The remainder is $= 2$ and the quotient is $= 2 {x}^{3} + 3 {x}^{2} + 2 x + 4$

Explanation:

Perform a long division

$\textcolor{w h i t e}{a a a a}$$4 {x}^{4} + 0 {x}^{3} - 5 {x}^{2} + 2 x - 10$$\textcolor{w h i t e}{a a a a}$$|$$2 x - 3$

$\textcolor{w h i t e}{a a a a}$$4 {x}^{4} - 6 {x}^{3}$$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a}$$|$$2 {x}^{3} + 3 {x}^{2} + 2 x + 4$

$\textcolor{w h i t e}{a a a a}$$0 {x}^{4} + 6 {x}^{3} - 5 {x}^{2}$

$\textcolor{w h i t e}{a a a a a a a a}$$+ 6 {x}^{3} - 9 {x}^{2}$

$\textcolor{w h i t e}{a a a a a a a a}$$+ 0 {x}^{3} + 4 {x}^{2} + 2 x$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$+ 4 {x}^{2} - 6 x$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$+ 0 {x}^{2} + 8 x - 10$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a}$$+ 8 x - 12$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a}$$+ 2$

The remainder is $= 2$ and the quotient is $= 2 {x}^{3} + 3 {x}^{2} + 2 x + 4$

$\frac{4 {x}^{4} + 0 {x}^{3} - 5 {x}^{2} + 2 x - 10}{2 x - 3} = \left(2 {x}^{3} + 3 {x}^{2} + 2 x + 4\right) + \frac{2}{2 x - 3}$