Algebra Examples

Use the long division method to find the result when $4 x^{3} + 10 x^{2} - 14 x - 15$ is divided by $2 x - 3$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{4 x^{3} + 10 x^{2} - 14 x - 15}{2 x - 3}$

Step 2

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

$2 x$

$3$

$4 x^{3}$

$10 x^{2}$

$14 x$

$15$

Step 3

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$

Step 4

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			+	$4 x^{3}$	-	$6 x^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 6 x^{2}$

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$

Step 6

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

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$

Step 7

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

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$

Step 8

Divide the highest order term in the dividend $16 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$8 x$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$

Step 9

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$8 x$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					+	$16 x^{2}$	-	$24 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{2} - 24 x$

				$2 x^{2}$	+	$8 x$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					-	$16 x^{2}$	+	$24 x$

Step 11

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

				$2 x^{2}$	+	$8 x$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					-	$16 x^{2}$	+	$24 x$
							+	$10 x$

Step 12

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

				$2 x^{2}$	+	$8 x$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					-	$16 x^{2}$	+	$24 x$
							+	$10 x$	-	$15$

Step 13

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$8 x$	+	$5$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					-	$16 x^{2}$	+	$24 x$
							+	$10 x$	-	$15$

Step 14

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$8 x$	+	$5$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					-	$16 x^{2}$	+	$24 x$
							+	$10 x$	-	$15$
							+	$10 x$	-	$15$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $10 x - 15$

				$2 x^{2}$	+	$8 x$	+	$5$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					-	$16 x^{2}$	+	$24 x$
							+	$10 x$	-	$15$
							-	$10 x$	+	$15$

Step 16

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

				$2 x^{2}$	+	$8 x$	+	$5$
$2 x$	-	$3$		$4 x^{3}$	+	$10 x^{2}$	-	$14 x$	-	$15$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$16 x^{2}$	-	$14 x$
					-	$16 x^{2}$	+	$24 x$
							+	$10 x$	-	$15$
							-	$10 x$	+	$15$
										$0$

Step 17

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

$2 x^{2} + 8 x + 5$