Algebra Examples

Use the long division method to find the result when $8 x^{3} + 24 x^{2} + 24 x + 7$ is divided by $2 x + 1$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{8 x^{3} + 24 x^{2} + 24 x + 7}{2 x + 1}$

Step 2

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

$2 x$

$1$

$8 x^{3}$

$24 x^{2}$

$24 x$

$7$

Step 3

Divide the highest order term in the dividend $8 x^{3}$ by the highest order term in divisor $2 x$ .

					$4 x^{2}$
	$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$

Step 4

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			+	$8 x^{3}$	+	$4 x^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} + 4 x^{2}$

				$4 x^{2}$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$

Step 6

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

				$4 x^{2}$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$

Step 7

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

				$4 x^{2}$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$

Step 8

Divide the highest order term in the dividend $20 x^{2}$ by the highest order term in divisor $2 x$ .

				$4 x^{2}$	+	$10 x$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$

Step 9

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$10 x$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					+	$20 x^{2}$	+	$10 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $20 x^{2} + 10 x$

				$4 x^{2}$	+	$10 x$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					-	$20 x^{2}$	-	$10 x$

Step 11

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

				$4 x^{2}$	+	$10 x$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					-	$20 x^{2}$	-	$10 x$
							+	$14 x$

Step 12

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

				$4 x^{2}$	+	$10 x$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					-	$20 x^{2}$	-	$10 x$
							+	$14 x$	+	$7$

Step 13

Divide the highest order term in the dividend $14 x$ by the highest order term in divisor $2 x$ .

				$4 x^{2}$	+	$10 x$	+	$7$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					-	$20 x^{2}$	-	$10 x$
							+	$14 x$	+	$7$

Step 14

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$10 x$	+	$7$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					-	$20 x^{2}$	-	$10 x$
							+	$14 x$	+	$7$
							+	$14 x$	+	$7$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $14 x + 7$

				$4 x^{2}$	+	$10 x$	+	$7$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					-	$20 x^{2}$	-	$10 x$
							+	$14 x$	+	$7$
							-	$14 x$	-	$7$

Step 16

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

				$4 x^{2}$	+	$10 x$	+	$7$
$2 x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$24 x$	+	$7$
			-	$8 x^{3}$	-	$4 x^{2}$
					+	$20 x^{2}$	+	$24 x$
					-	$20 x^{2}$	-	$10 x$
							+	$14 x$	+	$7$
							-	$14 x$	-	$7$
										$0$

Step 17

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

$4 x^{2} + 10 x + 7$