Precalculus Examples

$\frac{x^{3} - 2 x^{2} - 23 x + 60}{x - 4}$

Step 1

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

$x$

$4$

$x^{3}$

$2 x^{2}$

$23 x$

$60$

Step 2

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

					$x^{2}$
	$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			+	$x^{3}$	-	$4 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 4 x^{2}$

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$

Step 5

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$

Step 6

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$

Step 7

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

				$x^{2}$	+	$2 x$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					+	$2 x^{2}$	-	$8 x$

Step 9

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

				$x^{2}$	+	$2 x$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					-	$2 x^{2}$	+	$8 x$

Step 10

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

				$x^{2}$	+	$2 x$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					-	$2 x^{2}$	+	$8 x$
							-	$15 x$

Step 11

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

				$x^{2}$	+	$2 x$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					-	$2 x^{2}$	+	$8 x$
							-	$15 x$	+	$60$

Step 12

Divide the highest order term in the dividend $- 15 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					-	$2 x^{2}$	+	$8 x$
							-	$15 x$	+	$60$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					-	$2 x^{2}$	+	$8 x$
							-	$15 x$	+	$60$
							-	$15 x$	+	$60$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x + 60$

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					-	$2 x^{2}$	+	$8 x$
							-	$15 x$	+	$60$
							+	$15 x$	-	$60$

Step 15

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

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$4$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$4 x^{2}$
					+	$2 x^{2}$	-	$23 x$
					-	$2 x^{2}$	+	$8 x$
							-	$15 x$	+	$60$
							+	$15 x$	-	$60$
										$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 2 x - 15$