Algebra Examples

$\frac{x^{3} - 22 x + 15}{x + 5}$

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 $0$ .

$x$

$5$

$x^{3}$

$0 x^{2}$

$22 x$

$15$

Step 1.2

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

					$x^{2}$
	$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$

Step 1.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			+	$x^{3}$	+	$5 x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 5 x^{2}$

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$

Step 1.5

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$

Step 1.6

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$

Step 1.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$

Step 1.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					-	$5 x^{2}$	-	$25 x$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} - 25 x$

				$x^{2}$	-	$5 x$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	+	$25 x$

Step 1.10

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

				$x^{2}$	-	$5 x$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	+	$25 x$
							+	$3 x$

Step 1.11

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

				$x^{2}$	-	$5 x$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	+	$25 x$
							+	$3 x$	+	$15$

Step 1.12

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

				$x^{2}$	-	$5 x$	+	$3$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	+	$25 x$
							+	$3 x$	+	$15$

Step 1.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	+	$3$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	+	$25 x$
							+	$3 x$	+	$15$
							+	$3 x$	+	$15$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x + 15$

				$x^{2}$	-	$5 x$	+	$3$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	+	$25 x$
							+	$3 x$	+	$15$
							-	$3 x$	-	$15$

Step 1.15

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

				$x^{2}$	-	$5 x$	+	$3$
$x$	+	$5$		$x^{3}$	+	$0 x^{2}$	-	$22 x$	+	$15$
			-	$x^{3}$	-	$5 x^{2}$
					-	$5 x^{2}$	-	$22 x$
					+	$5 x^{2}$	+	$25 x$
							+	$3 x$	+	$15$
							-	$3 x$	-	$15$
										$0$

Step 1.16

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

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

Step 2

Since the final term in the resulting expression is not a fraction, the remainder is $0$ .

$0$