Divide Using Long Polynomial Division (8x^3-28x+61)/(2x+5)

$\frac{(8 x^{3} - 28 x + 61)}{2 x + 5}$

Step 1

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

$2 x$

$5$

$8 x^{3}$

$0 x^{2}$

$28 x$

$61$

Step 2

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

					$4 x^{2}$
	$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$

Step 3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			+	$8 x^{3}$	+	$20 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} + 20 x^{2}$

				$4 x^{2}$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$

Step 5

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

				$4 x^{2}$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$

Step 6

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

				$4 x^{2}$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$

Step 7

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

				$4 x^{2}$	-	$10 x$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$

Step 8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$10 x$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					-	$20 x^{2}$	-	$50 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x^{2} - 50 x$

				$4 x^{2}$	-	$10 x$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					+	$20 x^{2}$	+	$50 x$

Step 10

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

				$4 x^{2}$	-	$10 x$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					+	$20 x^{2}$	+	$50 x$
							+	$22 x$

Step 11

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

				$4 x^{2}$	-	$10 x$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					+	$20 x^{2}$	+	$50 x$
							+	$22 x$	+	$61$

Step 12

Divide the highest order term in the dividend $22 x$ by the highest order term in divisor $2 x$ .

				$4 x^{2}$	-	$10 x$	+	$11$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					+	$20 x^{2}$	+	$50 x$
							+	$22 x$	+	$61$

Step 13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$10 x$	+	$11$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					+	$20 x^{2}$	+	$50 x$
							+	$22 x$	+	$61$
							+	$22 x$	+	$55$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $22 x + 55$

				$4 x^{2}$	-	$10 x$	+	$11$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					+	$20 x^{2}$	+	$50 x$
							+	$22 x$	+	$61$
							-	$22 x$	-	$55$

Step 15

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

				$4 x^{2}$	-	$10 x$	+	$11$
$2 x$	+	$5$		$8 x^{3}$	+	$0 x^{2}$	-	$28 x$	+	$61$
			-	$8 x^{3}$	-	$20 x^{2}$
					-	$20 x^{2}$	-	$28 x$
					+	$20 x^{2}$	+	$50 x$
							+	$22 x$	+	$61$
							-	$22 x$	-	$55$
									+	$6$

Step 16

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

$4 x^{2} - 10 x + 11 + \frac{6}{2 x + 5}$

Divide Using Long Polynomial Division (8x^3-28x+61)/(2x+5)

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