Algebra Examples

$9 x^{3} - 12 x^{2} - 18 x - 9$ divided by $3 x - 7$

Step 1

Write the problem as a mathematical expression.

$\frac{9 x^{3} - 12 x^{2} - 18 x - 9}{3 x - 7}$

Step 2

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

$3 x$

$7$

$9 x^{3}$

$12 x^{2}$

$18 x$

$9$

Step 3

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

					$3 x^{2}$
	$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$

Step 4

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			+	$9 x^{3}$	-	$21 x^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} - 21 x^{2}$

				$3 x^{2}$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$

Step 6

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

				$3 x^{2}$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$

Step 7

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

				$3 x^{2}$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$

Step 8

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	+	$3 x$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$

Step 9

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$3 x$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$21 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} - 21 x$

				$3 x^{2}$	+	$3 x$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$21 x$

Step 11

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

				$3 x^{2}$	+	$3 x$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$21 x$
							+	$3 x$

Step 12

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

				$3 x^{2}$	+	$3 x$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$21 x$
							+	$3 x$	-	$9$

Step 13

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

				$3 x^{2}$	+	$3 x$	+	$1$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$21 x$
							+	$3 x$	-	$9$

Step 14

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$3 x$	+	$1$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$21 x$
							+	$3 x$	-	$9$
							+	$3 x$	-	$7$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $3 x - 7$

				$3 x^{2}$	+	$3 x$	+	$1$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$21 x$
							+	$3 x$	-	$9$
							-	$3 x$	+	$7$

Step 16

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

				$3 x^{2}$	+	$3 x$	+	$1$
$3 x$	-	$7$		$9 x^{3}$	-	$12 x^{2}$	-	$18 x$	-	$9$
			-	$9 x^{3}$	+	$21 x^{2}$
					+	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$21 x$
							+	$3 x$	-	$9$
							-	$3 x$	+	$7$
									-	$2$

Step 17

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

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