Pre-Algebra Examples

$(9 x^{3} + 12 x^{2} - 39 x - 10) \div (3 x - 5)$

Step 1

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

$3 x$

$5$

$9 x^{3}$

$12 x^{2}$

$39 x$

$10$

Step 2

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

					$3 x^{2}$
	$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$

Step 3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			+	$9 x^{3}$	-	$15 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} - 15 x^{2}$

				$3 x^{2}$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$

Step 5

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

				$3 x^{2}$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$

Step 6

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

				$3 x^{2}$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$

Step 7

Divide the highest order term in the dividend $27 x^{2}$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	+	$9 x$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$

Step 8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$9 x$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					+	$27 x^{2}$	-	$45 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $27 x^{2} - 45 x$

				$3 x^{2}$	+	$9 x$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					-	$27 x^{2}$	+	$45 x$

Step 10

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

				$3 x^{2}$	+	$9 x$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					-	$27 x^{2}$	+	$45 x$
							+	$6 x$

Step 11

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

				$3 x^{2}$	+	$9 x$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					-	$27 x^{2}$	+	$45 x$
							+	$6 x$	-	$10$

Step 12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	+	$9 x$	+	$2$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					-	$27 x^{2}$	+	$45 x$
							+	$6 x$	-	$10$

Step 13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$9 x$	+	$2$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					-	$27 x^{2}$	+	$45 x$
							+	$6 x$	-	$10$
							+	$6 x$	-	$10$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 10$

				$3 x^{2}$	+	$9 x$	+	$2$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					-	$27 x^{2}$	+	$45 x$
							+	$6 x$	-	$10$
							-	$6 x$	+	$10$

Step 15

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

				$3 x^{2}$	+	$9 x$	+	$2$
$3 x$	-	$5$		$9 x^{3}$	+	$12 x^{2}$	-	$39 x$	-	$10$
			-	$9 x^{3}$	+	$15 x^{2}$
					+	$27 x^{2}$	-	$39 x$
					-	$27 x^{2}$	+	$45 x$
							+	$6 x$	-	$10$
							-	$6 x$	+	$10$
										$0$

Step 16

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

$3 x^{2} + 9 x + 2$