Pre-Algebra Examples

$(x^{3} - 7 x - 6) \div (x - 2)$

Step 1

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

$x$

$2$

$x^{3}$

$0 x^{2}$

$7 x$

$6$

Step 2

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			+	$x^{3}$	-	$2 x^{2}$

Step 4

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$

Step 5

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$

Step 6

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$

Step 7

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

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					+	$2 x^{2}$	-	$4 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 4 x$

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					-	$2 x^{2}$	+	$4 x$

Step 10

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

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					-	$2 x^{2}$	+	$4 x$
							-	$3 x$

Step 11

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

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					-	$2 x^{2}$	+	$4 x$
							-	$3 x$	-	$6$

Step 12

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

				$x^{2}$	+	$2 x$	-	$3$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					-	$2 x^{2}$	+	$4 x$
							-	$3 x$	-	$6$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$3$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					-	$2 x^{2}$	+	$4 x$
							-	$3 x$	-	$6$
							-	$3 x$	+	$6$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x + 6$

				$x^{2}$	+	$2 x$	-	$3$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					-	$2 x^{2}$	+	$4 x$
							-	$3 x$	-	$6$
							+	$3 x$	-	$6$

Step 15

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

				$x^{2}$	+	$2 x$	-	$3$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$7 x$
					-	$2 x^{2}$	+	$4 x$
							-	$3 x$	-	$6$
							+	$3 x$	-	$6$
									-	$12$

Step 16

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

$x^{2} + 2 x - 3 - \frac{12}{x - 2}$