Algebra Examples

Use the long division method to find the result when $x^{3} - 6 x^{2} - 6 x + 20$ is divided by $x + 2$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{x^{3} - 6 x^{2} - 6 x + 20}{x + 2}$

Step 2

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

$x$

$2$

$x^{3}$

$6 x^{2}$

$6 x$

$20$

Step 3

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$

Step 4

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			+	$x^{3}$	+	$2 x^{2}$

Step 5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$

Step 6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$

Step 7

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$

Step 8

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$

Step 9

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					-	$8 x^{2}$	-	$16 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 16 x$

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					+	$8 x^{2}$	+	$16 x$

Step 11

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

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					+	$8 x^{2}$	+	$16 x$
							+	$10 x$

Step 12

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

				$x^{2}$	-	$8 x$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					+	$8 x^{2}$	+	$16 x$
							+	$10 x$	+	$20$

Step 13

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$	+	$10$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					+	$8 x^{2}$	+	$16 x$
							+	$10 x$	+	$20$

Step 14

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$10$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					+	$8 x^{2}$	+	$16 x$
							+	$10 x$	+	$20$
							+	$10 x$	+	$20$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $10 x + 20$

				$x^{2}$	-	$8 x$	+	$10$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					+	$8 x^{2}$	+	$16 x$
							+	$10 x$	+	$20$
							-	$10 x$	-	$20$

Step 16

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

				$x^{2}$	-	$8 x$	+	$10$
$x$	+	$2$		$x^{3}$	-	$6 x^{2}$	-	$6 x$	+	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$8 x^{2}$	-	$6 x$
					+	$8 x^{2}$	+	$16 x$
							+	$10 x$	+	$20$
							-	$10 x$	-	$20$
										$0$

Step 17

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

$x^{2} - 8 x + 10$