Algebra Examples

$\frac{9 x^{3} + 6 x^{2} + x + 22}{3 x + 4}$

Step 1

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

$3 x$

$4$

$9 x^{3}$

$6 x^{2}$

$x$

$22$

Step 2

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

					$3 x^{2}$
	$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$

Step 3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			+	$9 x^{3}$	+	$12 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} + 12 x^{2}$

				$3 x^{2}$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$

Step 5

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

				$3 x^{2}$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$

Step 6

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

				$3 x^{2}$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$

Step 7

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

				$3 x^{2}$	-	$2 x$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$

Step 8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$2 x$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$8 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 8 x$

				$3 x^{2}$	-	$2 x$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$8 x$

Step 10

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

				$3 x^{2}$	-	$2 x$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$8 x$
							+	$9 x$

Step 11

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

				$3 x^{2}$	-	$2 x$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$8 x$
							+	$9 x$	+	$22$

Step 12

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

				$3 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$8 x$
							+	$9 x$	+	$22$

Step 13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$8 x$
							+	$9 x$	+	$22$
							+	$9 x$	+	$12$

Step 14

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

				$3 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$8 x$
							+	$9 x$	+	$22$
							-	$9 x$	-	$12$

Step 15

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

				$3 x^{2}$	-	$2 x$	+	$3$
$3 x$	+	$4$		$9 x^{3}$	+	$6 x^{2}$	+	$x$	+	$22$
			-	$9 x^{3}$	-	$12 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$8 x$
							+	$9 x$	+	$22$
							-	$9 x$	-	$12$
									+	$10$

Step 16

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

$3 x^{2} - 2 x + 3 + \frac{10}{3 x + 4}$