Divide Using Long Polynomial Division (6r^3-23r^2-27r-40)/(r-5)

$\frac{6 r^{3} - 23 r^{2} - 27 r - 40}{r - 5}$

Step 1

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

$r$

$5$

$6 r^{3}$

$23 r^{2}$

$27 r$

$40$

Step 2

Divide the highest order term in the dividend $6 r^{3}$ by the highest order term in divisor $r$ .

					$6 r^{2}$
	$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$

Step 3

Multiply the new quotient term by the divisor.

				$6 r^{2}$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			+	$6 r^{3}$	-	$30 r^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 r^{3} - 30 r^{2}$

				$6 r^{2}$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$

Step 5

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

				$6 r^{2}$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$

Step 6

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

				$6 r^{2}$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$

Step 7

Divide the highest order term in the dividend $7 r^{2}$ by the highest order term in divisor $r$ .

				$6 r^{2}$	+	$7 r$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$

Step 8

Multiply the new quotient term by the divisor.

				$6 r^{2}$	+	$7 r$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					+	$7 r^{2}$	-	$35 r$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $7 r^{2} - 35 r$

				$6 r^{2}$	+	$7 r$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					-	$7 r^{2}$	+	$35 r$

Step 10

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

				$6 r^{2}$	+	$7 r$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					-	$7 r^{2}$	+	$35 r$
							+	$8 r$

Step 11

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

				$6 r^{2}$	+	$7 r$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					-	$7 r^{2}$	+	$35 r$
							+	$8 r$	-	$40$

Step 12

Divide the highest order term in the dividend $8 r$ by the highest order term in divisor $r$ .

				$6 r^{2}$	+	$7 r$	+	$8$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					-	$7 r^{2}$	+	$35 r$
							+	$8 r$	-	$40$

Step 13

Multiply the new quotient term by the divisor.

				$6 r^{2}$	+	$7 r$	+	$8$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					-	$7 r^{2}$	+	$35 r$
							+	$8 r$	-	$40$
							+	$8 r$	-	$40$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $8 r - 40$

				$6 r^{2}$	+	$7 r$	+	$8$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					-	$7 r^{2}$	+	$35 r$
							+	$8 r$	-	$40$
							-	$8 r$	+	$40$

Step 15

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

				$6 r^{2}$	+	$7 r$	+	$8$
$r$	-	$5$		$6 r^{3}$	-	$23 r^{2}$	-	$27 r$	-	$40$
			-	$6 r^{3}$	+	$30 r^{2}$
					+	$7 r^{2}$	-	$27 r$
					-	$7 r^{2}$	+	$35 r$
							+	$8 r$	-	$40$
							-	$8 r$	+	$40$
										$0$

Step 16

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

$6 r^{2} + 7 r + 8$

Divide Using Long Polynomial Division (6r^3-23r^2-27r-40)/(r-5)

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