Divide Using Long Polynomial Division (2r^3+5r^2-2r-15)÷(2r-3)

$(2 r^{3} + 5 r^{2} - 2 r - 15) \div (2 r - 3)$

Step 1

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

$2 r$

$3$

$2 r^{3}$

$5 r^{2}$

$2 r$

$15$

Step 2

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

					$r^{2}$
	$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$

Step 3

Multiply the new quotient term by the divisor.

				$r^{2}$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			+	$2 r^{3}$	-	$3 r^{2}$

Step 4

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

				$r^{2}$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$

Step 5

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

				$r^{2}$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$

Step 6

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

				$r^{2}$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$

Step 7

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

				$r^{2}$	+	$4 r$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$

Step 8

Multiply the new quotient term by the divisor.

				$r^{2}$	+	$4 r$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					+	$8 r^{2}$	-	$12 r$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $8 r^{2} - 12 r$

				$r^{2}$	+	$4 r$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					-	$8 r^{2}$	+	$12 r$

Step 10

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

				$r^{2}$	+	$4 r$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					-	$8 r^{2}$	+	$12 r$
							+	$10 r$

Step 11

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

				$r^{2}$	+	$4 r$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					-	$8 r^{2}$	+	$12 r$
							+	$10 r$	-	$15$

Step 12

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

				$r^{2}$	+	$4 r$	+	$5$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					-	$8 r^{2}$	+	$12 r$
							+	$10 r$	-	$15$

Step 13

Multiply the new quotient term by the divisor.

				$r^{2}$	+	$4 r$	+	$5$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					-	$8 r^{2}$	+	$12 r$
							+	$10 r$	-	$15$
							+	$10 r$	-	$15$

Step 14

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

				$r^{2}$	+	$4 r$	+	$5$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					-	$8 r^{2}$	+	$12 r$
							+	$10 r$	-	$15$
							-	$10 r$	+	$15$

Step 15

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

				$r^{2}$	+	$4 r$	+	$5$
$2 r$	-	$3$		$2 r^{3}$	+	$5 r^{2}$	-	$2 r$	-	$15$
			-	$2 r^{3}$	+	$3 r^{2}$
					+	$8 r^{2}$	-	$2 r$
					-	$8 r^{2}$	+	$12 r$
							+	$10 r$	-	$15$
							-	$10 r$	+	$15$
										$0$

Step 16

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

$r^{2} + 4 r + 5$

Divide Using Long Polynomial Division (2r^3+5r^2-2r-15)÷(2r-3)

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