Precalculus Examples

$\frac{x^{3} + 2 x^{2} - 2 x + 9}{x - 6}$

Step 1

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

$x$

$6$

$x^{3}$

$2 x^{2}$

$2 x$

$9$

Step 2

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

					$x^{2}$
	$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			+	$x^{3}$	-	$6 x^{2}$

Step 4

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$

Step 5

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$

Step 6

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$

Step 7

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	-	$48 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} - 48 x$

				$x^{2}$	+	$8 x$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	+	$48 x$

Step 10

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

				$x^{2}$	+	$8 x$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	+	$48 x$
							+	$46 x$

Step 11

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

				$x^{2}$	+	$8 x$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	+	$48 x$
							+	$46 x$	+	$9$

Step 12

Divide the highest order term in the dividend $46 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$	+	$46$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	+	$48 x$
							+	$46 x$	+	$9$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$	+	$46$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	+	$48 x$
							+	$46 x$	+	$9$
							+	$46 x$	-	$276$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $46 x - 276$

				$x^{2}$	+	$8 x$	+	$46$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	+	$48 x$
							+	$46 x$	+	$9$
							-	$46 x$	+	$276$

Step 15

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

				$x^{2}$	+	$8 x$	+	$46$
$x$	-	$6$		$x^{3}$	+	$2 x^{2}$	-	$2 x$	+	$9$
			-	$x^{3}$	+	$6 x^{2}$
					+	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	+	$48 x$
							+	$46 x$	+	$9$
							-	$46 x$	+	$276$
									+	$285$

Step 16

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

$x^{2} + 8 x + 46 + \frac{285}{x - 6}$