Algebra Examples

Use the long division method to find the result when $x^{3} + 6 x^{2} + 11 x + 6$ is divided by $x + 3$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{x^{3} + 6 x^{2} + 11 x + 6}{x + 3}$

Step 2

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

$x$

$3$

$x^{3}$

$6 x^{2}$

$11 x$

$6$

Step 3

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	+	$6 x^{2}$	+	$11 x$	+	$6$

Step 4

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$6 x^{2}$	+	$11 x$	+	$6$
			+	$x^{3}$	+	$3 x^{2}$

Step 5

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$6 x^{2}$	+	$11 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$

Step 6

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$6 x^{2}$	+	$11 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$3 x^{2}$

Step 7

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$6 x^{2}$	+	$11 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$3 x^{2}$	+	$11 x$

Step 8

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

				$x^{2}$	+	$3 x$
$x$	+	$3$		$x^{3}$	+	$6 x^{2}$	+	$11 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$3 x^{2}$	+	$11 x$

Step 9

Multiply the new quotient term by the divisor.

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

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

Step 14

Multiply the new quotient term by the divisor.

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

Step 15

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

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

Step 16

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

				$x^{2}$	+	$3 x$	+	$2$
$x$	+	$3$		$x^{3}$	+	$6 x^{2}$	+	$11 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$3 x^{2}$	+	$11 x$
					-	$3 x^{2}$	-	$9 x$
							+	$2 x$	+	$6$
							-	$2 x$	-	$6$
										$0$

Step 17

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

$x^{2} + 3 x + 2$