Pre-Algebra Examples

$\frac{32 x^{3} - 12 x^{2} - 51 x + 9}{4 x - 5}$

Step 1

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

$4 x$

$5$

$32 x^{3}$

$12 x^{2}$

$51 x$

$9$

Step 2

Divide the highest order term in the dividend $32 x^{3}$ by the highest order term in divisor $4 x$ .

					$8 x^{2}$
	$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$

Step 3

Multiply the new quotient term by the divisor.

				$8 x^{2}$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			+	$32 x^{3}$	-	$40 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $32 x^{3} - 40 x^{2}$

				$8 x^{2}$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$

Step 5

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

				$8 x^{2}$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$

Step 6

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

				$8 x^{2}$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$

Step 7

Divide the highest order term in the dividend $28 x^{2}$ by the highest order term in divisor $4 x$ .

				$8 x^{2}$	+	$7 x$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$

Step 8

Multiply the new quotient term by the divisor.

				$8 x^{2}$	+	$7 x$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					+	$28 x^{2}$	-	$35 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $28 x^{2} - 35 x$

				$8 x^{2}$	+	$7 x$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					-	$28 x^{2}$	+	$35 x$

Step 10

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

				$8 x^{2}$	+	$7 x$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					-	$28 x^{2}$	+	$35 x$
							-	$16 x$

Step 11

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

				$8 x^{2}$	+	$7 x$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					-	$28 x^{2}$	+	$35 x$
							-	$16 x$	+	$9$

Step 12

Divide the highest order term in the dividend $- 16 x$ by the highest order term in divisor $4 x$ .

				$8 x^{2}$	+	$7 x$	-	$4$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					-	$28 x^{2}$	+	$35 x$
							-	$16 x$	+	$9$

Step 13

Multiply the new quotient term by the divisor.

				$8 x^{2}$	+	$7 x$	-	$4$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					-	$28 x^{2}$	+	$35 x$
							-	$16 x$	+	$9$
							-	$16 x$	+	$20$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x + 20$

				$8 x^{2}$	+	$7 x$	-	$4$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					-	$28 x^{2}$	+	$35 x$
							-	$16 x$	+	$9$
							+	$16 x$	-	$20$

Step 15

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

				$8 x^{2}$	+	$7 x$	-	$4$
$4 x$	-	$5$		$32 x^{3}$	-	$12 x^{2}$	-	$51 x$	+	$9$
			-	$32 x^{3}$	+	$40 x^{2}$
					+	$28 x^{2}$	-	$51 x$
					-	$28 x^{2}$	+	$35 x$
							-	$16 x$	+	$9$
							+	$16 x$	-	$20$
									-	$11$

Step 16

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

$8 x^{2} + 7 x - 4 - \frac{11}{4 x - 5}$