Divide Using Long Polynomial Division (6x^4+2x^3-6x^2-14x-1)/(3x+1)

$\frac{6 x^{4} + 2 x^{3} - 6 x^{2} - 14 x - 1}{3 x + 1}$

Step 1

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

$3 x$

$1$

$6 x^{4}$

$2 x^{3}$

$6 x^{2}$

$14 x$

$1$

Step 2

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x$

$1$

$6 x^{4}$

$2 x^{3}$

$6 x^{2}$

$14 x$

$1$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x$

$1$

$6 x^{4}$

$2 x^{3}$

$6 x^{2}$

$14 x$

$1$

$6 x^{4}$

$2 x^{3}$

Step 4

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

				$2 x^{3}$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$

Step 5

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

				$2 x^{3}$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$

Step 6

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

				$2 x^{3}$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$

Step 7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $3 x$ .

				$2 x^{3}$	+	$0 x^{2}$	-	$2 x$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{3}$	+	$0 x^{2}$	-	$2 x$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$
							-	$6 x^{2}$	-	$2 x$

Step 9

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$3 x$

$1$

$6 x^{4}$

$2 x^{3}$

$6 x^{2}$

$14 x$

$1$

$6 x^{4}$

$2 x^{3}$

$0$

$6 x^{2}$

$14 x$

$6 x^{2}$

$2 x$

Step 10

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$3 x$

$1$

$6 x^{4}$

$2 x^{3}$

$6 x^{2}$

$14 x$

$1$

$6 x^{4}$

$2 x^{3}$

$0$

$6 x^{2}$

$14 x$

$6 x^{2}$

$2 x$

$12 x$

Step 11

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

				$2 x^{3}$	+	$0 x^{2}$	-	$2 x$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$
							+	$6 x^{2}$	+	$2 x$
									-	$12 x$	-	$1$

Step 12

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $3 x$ .

				$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$
							+	$6 x^{2}$	+	$2 x$
									-	$12 x$	-	$1$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$
							+	$6 x^{2}$	+	$2 x$
									-	$12 x$	-	$1$
									-	$12 x$	-	$4$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x - 4$

				$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$
							+	$6 x^{2}$	+	$2 x$
									-	$12 x$	-	$1$
									+	$12 x$	+	$4$

Step 15

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

				$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	-	$4$
$3 x$	+	$1$		$6 x^{4}$	+	$2 x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$1$
			-	$6 x^{4}$	-	$2 x^{3}$
						$0$	-	$6 x^{2}$	-	$14 x$
							+	$6 x^{2}$	+	$2 x$
									-	$12 x$	-	$1$
									+	$12 x$	+	$4$
											+	$3$

Step 16

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

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

Divide Using Long Polynomial Division (6x^4+2x^3-6x^2-14x-1)/(3x+1)

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