Examples

$\frac{x^{2} - x + 7}{x - 5}$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

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

$0$ .

$x$

$5$

$x^{2}$

$x$

$7$

Step 1.2

Divide the highest order term in the dividend

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

$x$ .

					$x$
	$x$	-	$5$		$x^{2}$	-	$x$	+	$7$

Step 1.3

Multiply the new quotient term by the divisor.

				$x$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			+	$x^{2}$	-	$5 x$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in

$x^{2} - 5 x$

				$x$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			-	$x^{2}$	+	$5 x$

Step 1.5

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

				$x$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			-	$x^{2}$	+	$5 x$
					+	$4 x$

Step 1.6

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

				$x$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			-	$x^{2}$	+	$5 x$
					+	$4 x$	+	$7$

Step 1.7

Divide the highest order term in the dividend

$4 x$ by the highest order term in divisor

$x$ .

				$x$	+	$4$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			-	$x^{2}$	+	$5 x$
					+	$4 x$	+	$7$

Step 1.8

Multiply the new quotient term by the divisor.

				$x$	+	$4$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			-	$x^{2}$	+	$5 x$
					+	$4 x$	+	$7$
					+	$4 x$	-	$20$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in

$4 x - 20$

				$x$	+	$4$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			-	$x^{2}$	+	$5 x$
					+	$4 x$	+	$7$
					-	$4 x$	+	$20$

Step 1.10

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

				$x$	+	$4$
$x$	-	$5$		$x^{2}$	-	$x$	+	$7$
			-	$x^{2}$	+	$5 x$
					+	$4 x$	+	$7$
					-	$4 x$	+	$20$
							+	$27$

Step 1.11

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

$x + 4 + \frac{27}{x - 5}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$27$

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