Pre-Algebra Examples

$\frac{12 x^{3} - 2 x^{2} + 7 x - 5}{3 x + 1}$

Step 1

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

$3 x$

$1$

$12 x^{3}$

$2 x^{2}$

$7 x$

$5$

Step 2

Divide the highest order term in the dividend $12 x^{3}$ by the highest order term in divisor $3 x$ .

					$4 x^{2}$
	$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$

Step 3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			+	$12 x^{3}$	+	$4 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{3} + 4 x^{2}$

				$4 x^{2}$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$

Step 5

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

				$4 x^{2}$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$

Step 6

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

				$4 x^{2}$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$

Step 7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $3 x$ .

				$4 x^{2}$	-	$2 x$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$

Step 8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$2 x$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					-	$6 x^{2}$	-	$2 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 2 x$

				$4 x^{2}$	-	$2 x$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$2 x$

Step 10

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

				$4 x^{2}$	-	$2 x$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$2 x$
							+	$9 x$

Step 11

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

				$4 x^{2}$	-	$2 x$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$2 x$
							+	$9 x$	-	$5$

Step 12

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $3 x$ .

				$4 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$2 x$
							+	$9 x$	-	$5$

Step 13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$2 x$
							+	$9 x$	-	$5$
							+	$9 x$	+	$3$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $9 x + 3$

				$4 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$2 x$
							+	$9 x$	-	$5$
							-	$9 x$	-	$3$

Step 15

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

				$4 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$1$		$12 x^{3}$	-	$2 x^{2}$	+	$7 x$	-	$5$
			-	$12 x^{3}$	-	$4 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$2 x$
							+	$9 x$	-	$5$
							-	$9 x$	-	$3$
									-	$8$

Step 16

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

$4 x^{2} - 2 x + 3 - \frac{8}{3 x + 1}$