Pre-Algebra Examples

$(x^{3} - 4 x^{2} - 17 x + 6) \div (x - 3)$

Step 1

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

$x$

$3$

$x^{3}$

$4 x^{2}$

$17 x$

$6$

Step 2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$

Step 3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			+	$x^{3}$	-	$3 x^{2}$

Step 4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$

Step 5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$

Step 6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$

Step 7

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

				$x^{2}$	-	$x$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$

Step 8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$3 x$

Step 9

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

				$x^{2}$	-	$x$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$3 x$

Step 10

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

				$x^{2}$	-	$x$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$3 x$
							-	$20 x$

Step 11

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

				$x^{2}$	-	$x$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$3 x$
							-	$20 x$	+	$6$

Step 12

Divide the highest order term in the dividend $- 20 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$3 x$
							-	$20 x$	+	$6$

Step 13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$3 x$
							-	$20 x$	+	$6$
							-	$20 x$	+	$60$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x + 60$

				$x^{2}$	-	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$3 x$
							-	$20 x$	+	$6$
							+	$20 x$	-	$60$

Step 15

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

				$x^{2}$	-	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	-	$17 x$	+	$6$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$3 x$
							-	$20 x$	+	$6$
							+	$20 x$	-	$60$
									-	$54$

Step 16

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

$x^{2} - x - 20 - \frac{54}{x - 3}$