Pre-Algebra Examples

$\frac{- 6 x^{3} + 18 x^{2} - 7 x - 10}{x - 2}$

Step 1

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

$x$

$2$

$6 x^{3}$

$18 x^{2}$

$7 x$

$10$

Step 2

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

				-	$6 x^{2}$
	$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$

Step 3

Multiply the new quotient term by the divisor.

			-	$6 x^{2}$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			-	$6 x^{3}$	+	$12 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{3} + 12 x^{2}$

			-	$6 x^{2}$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$

Step 5

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

			-	$6 x^{2}$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$

Step 6

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

			-	$6 x^{2}$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$

Step 7

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

			-	$6 x^{2}$	+	$6 x$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$

Step 8

Multiply the new quotient term by the divisor.

			-	$6 x^{2}$	+	$6 x$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					+	$6 x^{2}$	-	$12 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 12 x$

			-	$6 x^{2}$	+	$6 x$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$12 x$

Step 10

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

			-	$6 x^{2}$	+	$6 x$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$12 x$
							+	$5 x$

Step 11

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

			-	$6 x^{2}$	+	$6 x$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$12 x$
							+	$5 x$	-	$10$

Step 12

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $x$ .

			-	$6 x^{2}$	+	$6 x$	+	$5$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$12 x$
							+	$5 x$	-	$10$

Step 13

Multiply the new quotient term by the divisor.

			-	$6 x^{2}$	+	$6 x$	+	$5$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$12 x$
							+	$5 x$	-	$10$
							+	$5 x$	-	$10$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $5 x - 10$

			-	$6 x^{2}$	+	$6 x$	+	$5$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$12 x$
							+	$5 x$	-	$10$
							-	$5 x$	+	$10$

Step 15

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

			-	$6 x^{2}$	+	$6 x$	+	$5$
$x$	-	$2$	-	$6 x^{3}$	+	$18 x^{2}$	-	$7 x$	-	$10$
			+	$6 x^{3}$	-	$12 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$12 x$
							+	$5 x$	-	$10$
							-	$5 x$	+	$10$
										$0$

Step 16

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

$- 6 x^{2} + 6 x + 5$