Pre-Algebra Examples

$\frac{x^{3} + 8 x^{2} + 10 x - 23}{x^{2} + 3 x - 4}$

Step 1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

Step 2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

Step 3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

$x^{3}$

$3 x^{2}$

$4 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 3 x^{2} - 4 x$

$x$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

$x^{3}$

$3 x^{2}$

$4 x$

Step 5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

$x^{3}$

$3 x^{2}$

$4 x$

$5 x^{2}$

$14 x$

Step 6

Pull the next terms from the original dividend down into the current dividend.

$x$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

$x^{3}$

$3 x^{2}$

$4 x$

$5 x^{2}$

$14 x$

$23$

Step 7

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x$

$5$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

$x^{3}$

$3 x^{2}$

$4 x$

$5 x^{2}$

$14 x$

$23$

Step 8

Multiply the new quotient term by the divisor.

$x$

$5$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

$x^{3}$

$3 x^{2}$

$4 x$

$5 x^{2}$

$14 x$

$23$

$5 x^{2}$

$15 x$

$20$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 15 x - 20$

$x$

$5$

$x^{2}$

$3 x$

$4$

$x^{3}$

$8 x^{2}$

$10 x$

$23$

$x^{3}$

$3 x^{2}$

$4 x$

$5 x^{2}$

$14 x$

$23$

$5 x^{2}$

$15 x$

$20$

Step 10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

						$x$	+	$5$
$x^{2}$	+	$3 x$	-	$4$		$x^{3}$	+	$8 x^{2}$	+	$10 x$	-	$23$
					-	$x^{3}$	-	$3 x^{2}$	+	$4 x$
							+	$5 x^{2}$	+	$14 x$	-	$23$
							-	$5 x^{2}$	-	$15 x$	+	$20$
									-	$x$	-	$3$

Step 11

The final answer is the quotient plus the remainder over the divisor.

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