Pre-Algebra Examples

$\frac{2 x^{3} + 4 x^{2} - 5 x + 16}{x - 3}$

Step 1

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

$x$

$3$

$2 x^{3}$

$4 x^{2}$

$5 x$

$16$

Step 2

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

					$2 x^{2}$
	$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$

Step 3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			+	$2 x^{3}$	-	$6 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 6 x^{2}$

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$

Step 5

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$

Step 6

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$

Step 7

Divide the highest order term in the dividend $10 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$10 x$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$10 x$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					+	$10 x^{2}$	-	$30 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{2} - 30 x$

				$2 x^{2}$	+	$10 x$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					-	$10 x^{2}$	+	$30 x$

Step 10

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

				$2 x^{2}$	+	$10 x$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					-	$10 x^{2}$	+	$30 x$
							+	$25 x$

Step 11

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

				$2 x^{2}$	+	$10 x$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					-	$10 x^{2}$	+	$30 x$
							+	$25 x$	+	$16$

Step 12

Divide the highest order term in the dividend $25 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$10 x$	+	$25$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					-	$10 x^{2}$	+	$30 x$
							+	$25 x$	+	$16$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$10 x$	+	$25$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					-	$10 x^{2}$	+	$30 x$
							+	$25 x$	+	$16$
							+	$25 x$	-	$75$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $25 x - 75$

				$2 x^{2}$	+	$10 x$	+	$25$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					-	$10 x^{2}$	+	$30 x$
							+	$25 x$	+	$16$
							-	$25 x$	+	$75$

Step 15

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

				$2 x^{2}$	+	$10 x$	+	$25$
$x$	-	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$5 x$	+	$16$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$10 x^{2}$	-	$5 x$
					-	$10 x^{2}$	+	$30 x$
							+	$25 x$	+	$16$
							-	$25 x$	+	$75$
									+	$91$

Step 16

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

$2 x^{2} + 10 x + 25 + \frac{91}{x - 3}$