Divide Using Long Polynomial Division (4x^3+5x+10)/(2x+3)

$\frac{4 x^{3} + 5 x + 10}{2 x + 3}$

Step 1

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

$2 x$

$3$

$4 x^{3}$

$0 x^{2}$

$5 x$

$10$

Step 2

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

					$2 x^{2}$
	$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$

Step 3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			+	$4 x^{3}$	+	$6 x^{2}$

Step 4

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

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$

Step 5

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

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$

Step 6

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

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$

Step 7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					-	$6 x^{2}$	-	$9 x$

Step 9

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

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					+	$6 x^{2}$	+	$9 x$

Step 10

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

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					+	$6 x^{2}$	+	$9 x$
							+	$14 x$

Step 11

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

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					+	$6 x^{2}$	+	$9 x$
							+	$14 x$	+	$10$

Step 12

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

				$2 x^{2}$	-	$3 x$	+	$7$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					+	$6 x^{2}$	+	$9 x$
							+	$14 x$	+	$10$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$	+	$7$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					+	$6 x^{2}$	+	$9 x$
							+	$14 x$	+	$10$
							+	$14 x$	+	$21$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $14 x + 21$

				$2 x^{2}$	-	$3 x$	+	$7$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					+	$6 x^{2}$	+	$9 x$
							+	$14 x$	+	$10$
							-	$14 x$	-	$21$

Step 15

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

				$2 x^{2}$	-	$3 x$	+	$7$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$10$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$5 x$
					+	$6 x^{2}$	+	$9 x$
							+	$14 x$	+	$10$
							-	$14 x$	-	$21$
									-	$11$

Step 16

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

$2 x^{2} - 3 x + 7 - \frac{11}{2 x + 3}$

Divide Using Long Polynomial Division (4x^3+5x+10)/(2x+3)

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