Divide Using Long Polynomial Division (12a^3-6a^2-2a+15)/(4a+2)

$\frac{12 a^{3} - 6 a^{2} - 2 a + 15}{4 a + 2}$

Step 1

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

$4 a$

$2$

$12 a^{3}$

$6 a^{2}$

$2 a$

$15$

Step 2

Divide the highest order term in the dividend $12 a^{3}$ by the highest order term in divisor $4 a$ .

					$3 a^{2}$
	$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$

Step 3

Multiply the new quotient term by the divisor.

				$3 a^{2}$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			+	$12 a^{3}$	+	$6 a^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $12 a^{3} + 6 a^{2}$

				$3 a^{2}$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$

Step 5

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

				$3 a^{2}$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$

Step 6

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

				$3 a^{2}$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$

Step 7

Divide the highest order term in the dividend $- 12 a^{2}$ by the highest order term in divisor $4 a$ .

				$3 a^{2}$	-	$3 a$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$

Step 8

Multiply the new quotient term by the divisor.

				$3 a^{2}$	-	$3 a$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					-	$12 a^{2}$	-	$6 a$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 a^{2} - 6 a$

				$3 a^{2}$	-	$3 a$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					+	$12 a^{2}$	+	$6 a$

Step 10

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

				$3 a^{2}$	-	$3 a$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					+	$12 a^{2}$	+	$6 a$
							+	$4 a$

Step 11

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

				$3 a^{2}$	-	$3 a$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					+	$12 a^{2}$	+	$6 a$
							+	$4 a$	+	$15$

Step 12

Divide the highest order term in the dividend $4 a$ by the highest order term in divisor $4 a$ .

				$3 a^{2}$	-	$3 a$	+	$1$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					+	$12 a^{2}$	+	$6 a$
							+	$4 a$	+	$15$

Step 13

Multiply the new quotient term by the divisor.

				$3 a^{2}$	-	$3 a$	+	$1$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					+	$12 a^{2}$	+	$6 a$
							+	$4 a$	+	$15$
							+	$4 a$	+	$2$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $4 a + 2$

				$3 a^{2}$	-	$3 a$	+	$1$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					+	$12 a^{2}$	+	$6 a$
							+	$4 a$	+	$15$
							-	$4 a$	-	$2$

Step 15

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

				$3 a^{2}$	-	$3 a$	+	$1$
$4 a$	+	$2$		$12 a^{3}$	-	$6 a^{2}$	-	$2 a$	+	$15$
			-	$12 a^{3}$	-	$6 a^{2}$
					-	$12 a^{2}$	-	$2 a$
					+	$12 a^{2}$	+	$6 a$
							+	$4 a$	+	$15$
							-	$4 a$	-	$2$
									+	$13$

Step 16

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

$3 a^{2} - 3 a + 1 + \frac{13}{4 a + 2}$

Divide Using Long Polynomial Division (12a^3-6a^2-2a+15)/(4a+2)

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