Precalculus Examples

$\frac{x^{3} - 2 x^{2} + 5 x + 1}{x - 1}$

Step 1

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

$x$

$1$

$x^{3}$

$2 x^{2}$

$5 x$

$1$

Step 2

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			+	$x^{3}$	-	$x^{2}$

Step 4

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$

Step 5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$

Step 6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$

Step 7

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					-	$x^{2}$	+	$x$

Step 9

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	-	$x$

Step 10

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	-	$x$
							+	$4 x$

Step 11

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	-	$x$
							+	$4 x$	+	$1$

Step 12

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

				$x^{2}$	-	$x$	+	$4$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	-	$x$
							+	$4 x$	+	$1$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	+	$4$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	-	$x$
							+	$4 x$	+	$1$
							+	$4 x$	-	$4$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 4$

				$x^{2}$	-	$x$	+	$4$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	-	$x$
							+	$4 x$	+	$1$
							-	$4 x$	+	$4$

Step 15

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

				$x^{2}$	-	$x$	+	$4$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$5 x$	+	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	-	$x$
							+	$4 x$	+	$1$
							-	$4 x$	+	$4$
									+	$5$

Step 16

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

$x^{2} - x + 4 + \frac{5}{x - 1}$