Algebra Examples

Use the long division method to find the result when $3 x^{3} - 11 x^{2} - 18 x + 16$ is divided by $3 x - 2$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{3 x^{3} - 11 x^{2} - 18 x + 16}{3 x - 2}$

Step 2

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

$3 x$

$2$

$3 x^{3}$

$11 x^{2}$

$18 x$

$16$

Step 3

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

					$x^{2}$
	$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$

Step 4

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			+	$3 x^{3}$	-	$2 x^{2}$

Step 5

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

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$

Step 6

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

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$

Step 7

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

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$

Step 8

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

				$x^{2}$	-	$3 x$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$

Step 9

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					-	$9 x^{2}$	+	$6 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{2} + 6 x$

				$x^{2}$	-	$3 x$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$6 x$

Step 11

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

				$x^{2}$	-	$3 x$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$6 x$
							-	$24 x$

Step 12

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

				$x^{2}$	-	$3 x$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$6 x$
							-	$24 x$	+	$16$

Step 13

Divide the highest order term in the dividend $- 24 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	-	$3 x$	-	$8$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$6 x$
							-	$24 x$	+	$16$

Step 14

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	-	$8$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$6 x$
							-	$24 x$	+	$16$
							-	$24 x$	+	$16$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x + 16$

				$x^{2}$	-	$3 x$	-	$8$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$6 x$
							-	$24 x$	+	$16$
							+	$24 x$	-	$16$

Step 16

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

				$x^{2}$	-	$3 x$	-	$8$
$3 x$	-	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$18 x$	+	$16$
			-	$3 x^{3}$	+	$2 x^{2}$
					-	$9 x^{2}$	-	$18 x$
					+	$9 x^{2}$	-	$6 x$
							-	$24 x$	+	$16$
							+	$24 x$	-	$16$
										$0$

Step 17

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

$x^{2} - 3 x - 8$