How do you divide (2x^4 + 5x^3 - 2x^2 + 5x + 3)/(x-1)?

Apr 24, 2017

Long division or synthetic division
$2 {x}^{3} + 7 {x}^{2} + 5 x + 10 + \frac{13}{x - 1}$

Explanation:

Given:$\frac{2 {x}^{4} + 5 {x}^{3} - 2 {x}^{2} + 5 x + 3}{x - 1}$

Long Division:

$\text{ } 2 {x}^{3} + 7 {x}^{2} + 5 x + 10 + \frac{13}{x - 1}$
$x - 1 | \overline{2 {x}^{4} + 5 {x}^{3} - 2 {x}^{2} + 5 x + 3}$
$\text{ } \underline{2 {x}^{4} - 2 {x}^{3}}$
$\text{ } 7 {x}^{3} - 2 {x}^{2}$
$\text{ } \underline{7 {x}^{3} - 7 {x}^{2}}$
$\text{ } 5 {x}^{2} + 5 x$
$\text{ } \underline{5 {x}^{2} - 5 x}$
$\text{ } 10 x + 3$
$\text{ } \underline{10 x - 10}$
$\text{ } 13$

Synthetic Division , where $x - 1 = 0 \text{ or } x = 1 :$

terms:$\text{ "x^4" "x^3" "x^2" "x" constant}$

$\underline{1} | \text{ "2" "5" "-2" "5" } 3$
" "ul(+" "2" "7" "5" "10)
$\text{ "2" "7" "5" "10" } 13$

terms:$\text{ "x^3" "x^2" "x" constant, remainder}$

This means

$\frac{2 {x}^{4} + 5 {x}^{3} - 2 {x}^{2} + 5 x + 3}{x - 1} = 2 {x}^{3} + 7 {x}^{2} + 5 x + 10 + \frac{13}{x - 1}$