Pre-Algebra Examples

$(20 x^{3} - 27 x^{2} + x + 6) \div (4 x - 3)$

Step 1

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

$4 x$

$3$

$20 x^{3}$

$27 x^{2}$

$x$

$6$

Step 2

Divide the highest order term in the dividend $20 x^{3}$ by the highest order term in divisor $4 x$ .

					$5 x^{2}$
	$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$

Step 3

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			+	$20 x^{3}$	-	$15 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $20 x^{3} - 15 x^{2}$

				$5 x^{2}$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$

Step 5

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

				$5 x^{2}$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$

Step 6

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

				$5 x^{2}$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$

Step 7

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $4 x$ .

				$5 x^{2}$	-	$3 x$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$

Step 8

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$3 x$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					-	$12 x^{2}$	+	$9 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} + 9 x$

				$5 x^{2}$	-	$3 x$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	-	$9 x$

Step 10

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

				$5 x^{2}$	-	$3 x$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	-	$9 x$
							-	$8 x$

Step 11

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

				$5 x^{2}$	-	$3 x$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	-	$9 x$
							-	$8 x$	+	$6$

Step 12

Divide the highest order term in the dividend $- 8 x$ by the highest order term in divisor $4 x$ .

				$5 x^{2}$	-	$3 x$	-	$2$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	-	$9 x$
							-	$8 x$	+	$6$

Step 13

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$3 x$	-	$2$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	-	$9 x$
							-	$8 x$	+	$6$
							-	$8 x$	+	$6$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x + 6$

				$5 x^{2}$	-	$3 x$	-	$2$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	-	$9 x$
							-	$8 x$	+	$6$
							+	$8 x$	-	$6$

Step 15

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

				$5 x^{2}$	-	$3 x$	-	$2$
$4 x$	-	$3$		$20 x^{3}$	-	$27 x^{2}$	+	$x$	+	$6$
			-	$20 x^{3}$	+	$15 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	-	$9 x$
							-	$8 x$	+	$6$
							+	$8 x$	-	$6$
										$0$

Step 16

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

$5 x^{2} - 3 x - 2$