Precalculus Examples

$\frac{5 x^{3} + 6 x + 8}{x + 2}$

Step 1

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

$x$

$2$

$5 x^{3}$

$0 x^{2}$

$6 x$

$8$

Step 2

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

					$5 x^{2}$
	$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$

Step 3

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			+	$5 x^{3}$	+	$10 x^{2}$

Step 4

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

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$

Step 5

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

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$

Step 6

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

				$5 x^{2}$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$

Step 7

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

				$5 x^{2}$	-	$10 x$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$

Step 8

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$10 x$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					-	$10 x^{2}$	-	$20 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 20 x$

				$5 x^{2}$	-	$10 x$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					+	$10 x^{2}$	+	$20 x$

Step 10

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

				$5 x^{2}$	-	$10 x$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					+	$10 x^{2}$	+	$20 x$
							+	$26 x$

Step 11

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

				$5 x^{2}$	-	$10 x$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					+	$10 x^{2}$	+	$20 x$
							+	$26 x$	+	$8$

Step 12

Divide the highest order term in the dividend $26 x$ by the highest order term in divisor $x$ .

				$5 x^{2}$	-	$10 x$	+	$26$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					+	$10 x^{2}$	+	$20 x$
							+	$26 x$	+	$8$

Step 13

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$10 x$	+	$26$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					+	$10 x^{2}$	+	$20 x$
							+	$26 x$	+	$8$
							+	$26 x$	+	$52$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $26 x + 52$

				$5 x^{2}$	-	$10 x$	+	$26$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					+	$10 x^{2}$	+	$20 x$
							+	$26 x$	+	$8$
							-	$26 x$	-	$52$

Step 15

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

				$5 x^{2}$	-	$10 x$	+	$26$
$x$	+	$2$		$5 x^{3}$	+	$0 x^{2}$	+	$6 x$	+	$8$
			-	$5 x^{3}$	-	$10 x^{2}$
					-	$10 x^{2}$	+	$6 x$
					+	$10 x^{2}$	+	$20 x$
							+	$26 x$	+	$8$
							-	$26 x$	-	$52$
									-	$44$

Step 16

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

$5 x^{2} - 10 x + 26 - \frac{44}{x + 2}$