Precalculus Examples

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

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

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

$0$ .

x

$x$

4

$4$

x^{2}

$x^{2}$

5 x

$5 x$

0

$0$

Step 1.2

Divide the highest order term in the dividend

x^{2}

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

x

$x$ .

					$x$ $x$
	$x$ $x$	-	$4$ $4$		$x^{2}$ $x^{2}$	+	$5 x$ $5 x$	+	$0$ $0$

Step 1.3

Multiply the new quotient term by the divisor.

				$x$ $x$
$x$ $x$	-	$4$ $4$		$x^{2}$ $x^{2}$	+	$5 x$ $5 x$	+	$0$ $0$
			+	$x^{2}$ $x^{2}$	-	$4 x$ $4 x$

Step 1.4

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

x^{2} - 4 x

$x^{2} - 4 x$

				$x$ $x$
$x$ $x$	-	$4$ $4$		$x^{2}$ $x^{2}$	+	$5 x$ $5 x$	+	$0$ $0$
			-	$x^{2}$ $x^{2}$	+	$4 x$ $4 x$

Step 1.5

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

				$x$ $x$
$x$ $x$	-	$4$ $4$		$x^{2}$ $x^{2}$	+	$5 x$ $5 x$	+	$0$ $0$
			-	$x^{2}$ $x^{2}$	+	$4 x$ $4 x$
					+	$9 x$ $9 x$

Step 1.6

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

				$x$ $x$
$x$ $x$	-	$4$ $4$		$x^{2}$ $x^{2}$	+	$5 x$ $5 x$	+	$0$ $0$
			-	$x^{2}$ $x^{2}$	+	$4 x$ $4 x$
					+	$9 x$ $9 x$	+	$0$ $0$

Step 1.7

Divide the highest order term in the dividend

9 x

$9 x$ by the highest order term in divisor

x

$x$ .

				$x$ $x$	+	$9$ $9$
$x$ $x$	-	$4$ $4$		$x^{2}$ $x^{2}$	+	$5 x$ $5 x$	+	$0$ $0$
			-	$x^{2}$ $x^{2}$	+	$4 x$ $4 x$
					+	$9 x$ $9 x$	+	$0$ $0$

Step 1.8

Multiply the new quotient term by the divisor.

				$x$ $x$	+	$9$ $9$
$x$ $x$	-	$4$ $4$		$x^{2}$	+	$5 x$	+	$0$
			-	$x^{2}$	+	$4 x$
					+	$9 x$	+	$0$
					+	$9 x$	-	$36$

Step 1.9

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

$9 x - 36$

				$x$	+	$9$
$x$	-	$4$		$x^{2}$	+	$5 x$	+	$0$
			-	$x^{2}$	+	$4 x$
					+	$9 x$	+	$0$
					-	$9 x$	+	$36$

Step 1.10

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

				$x$	+	$9$
$x$	-	$4$		$x^{2}$	+	$5 x$	+	$0$
			-	$x^{2}$	+	$4 x$
					+	$9 x$	+	$0$
					-	$9 x$	+	$36$
							+	$36$

Step 1.11

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

$x + 9 + \frac{36}{x - 4}$

Step 2

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

$36$

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