Pre-Algebra Examples

$\frac{x^{3} + 4 x - 7}{x - 3}$

Step 1

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

$x$

$3$

$x^{3}$

$0 x^{2}$

$4 x$

$7$

Step 2

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

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

Step 3

Multiply the new quotient term by the divisor.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

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

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - 9 x$

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$

Step 10

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

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$
							+	$13 x$

Step 11

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

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$
							+	$13 x$	-	$7$

Step 12

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	+	$13$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$
							+	$13 x$	-	$7$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	+	$13$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$
							+	$13 x$	-	$7$
							+	$13 x$	-	$39$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $13 x - 39$

				$x^{2}$	+	$3 x$	+	$13$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$
							+	$13 x$	-	$7$
							-	$13 x$	+	$39$

Step 15

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

				$x^{2}$	+	$3 x$	+	$13$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$7$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$
							+	$13 x$	-	$7$
							-	$13 x$	+	$39$
									+	$32$

Step 16

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

$x^{2} + 3 x + 13 + \frac{32}{x - 3}$