Pre-Algebra Examples

$(5 x^{3} - 12 x^{2} - 36 x - 16) \div (x + 2)$

Step 1

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

$x$

$2$

$5 x^{3}$

$12 x^{2}$

$36 x$

$16$

Step 2

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

					$5 x^{2}$
	$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$

Step 3

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			+	$5 x^{3}$	+	$10 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 10 x^{2}$

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$

Step 5

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

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$

Step 6

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

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$

Step 7

Divide the highest order term in the dividend $- 22 x^{2}$ by the highest order term in divisor $x$ .

				$5 x^{2}$	-	$22 x$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$

Step 8

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$22 x$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					-	$22 x^{2}$	-	$44 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 22 x^{2} - 44 x$

				$5 x^{2}$	-	$22 x$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					+	$22 x^{2}$	+	$44 x$

Step 10

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

				$5 x^{2}$	-	$22 x$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					+	$22 x^{2}$	+	$44 x$
							+	$8 x$

Step 11

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

				$5 x^{2}$	-	$22 x$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					+	$22 x^{2}$	+	$44 x$
							+	$8 x$	-	$16$

Step 12

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $x$ .

				$5 x^{2}$	-	$22 x$	+	$8$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					+	$22 x^{2}$	+	$44 x$
							+	$8 x$	-	$16$

Step 13

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$22 x$	+	$8$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					+	$22 x^{2}$	+	$44 x$
							+	$8 x$	-	$16$
							+	$8 x$	+	$16$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $8 x + 16$

				$5 x^{2}$	-	$22 x$	+	$8$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					+	$22 x^{2}$	+	$44 x$
							+	$8 x$	-	$16$
							-	$8 x$	-	$16$

Step 15

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

				$5 x^{2}$	-	$22 x$	+	$8$
$x$	+	$2$		$5 x^{3}$	-	$12 x^{2}$	-	$36 x$	-	$16$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$22 x^{2}$	-	$36 x$
					+	$22 x^{2}$	+	$44 x$
							+	$8 x$	-	$16$
							-	$8 x$	-	$16$
									-	$32$

Step 16

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

$5 x^{2} - 22 x + 8 - \frac{32}{x + 2}$