Divide Using Long Polynomial Division Use the long division method to find the result when 6x^3+5x^2+19x+6 is divided by 3x+1

Use the long division method to find the result when $6 x^{3} + 5 x^{2} + 19 x + 6$ is divided by $3 x + 1$

Step 1

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{6 x^{3} + 5 x^{2} + 19 x + 6}{3 x + 1}$

Step 2

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

$3 x$

$1$

$6 x^{3}$

$5 x^{2}$

$19 x$

$6$

Step 3

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

					$2 x^{2}$
	$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$

Step 4

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			+	$6 x^{3}$	+	$2 x^{2}$

Step 5

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

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$

Step 6

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

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$

Step 7

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

				$2 x^{2}$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$

Step 8

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

				$2 x^{2}$	+	$x$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$

Step 9

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					+	$3 x^{2}$	+	$x$

Step 10

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

				$2 x^{2}$	+	$x$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					-	$3 x^{2}$	-	$x$

Step 11

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

				$2 x^{2}$	+	$x$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					-	$3 x^{2}$	-	$x$
							+	$18 x$

Step 12

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

				$2 x^{2}$	+	$x$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					-	$3 x^{2}$	-	$x$
							+	$18 x$	+	$6$

Step 13

Divide the highest order term in the dividend $18 x$ by the highest order term in divisor $3 x$ .

				$2 x^{2}$	+	$x$	+	$6$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					-	$3 x^{2}$	-	$x$
							+	$18 x$	+	$6$

Step 14

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$	+	$6$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					-	$3 x^{2}$	-	$x$
							+	$18 x$	+	$6$
							+	$18 x$	+	$6$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $18 x + 6$

				$2 x^{2}$	+	$x$	+	$6$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					-	$3 x^{2}$	-	$x$
							+	$18 x$	+	$6$
							-	$18 x$	-	$6$

Step 16

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

				$2 x^{2}$	+	$x$	+	$6$
$3 x$	+	$1$		$6 x^{3}$	+	$5 x^{2}$	+	$19 x$	+	$6$
			-	$6 x^{3}$	-	$2 x^{2}$
					+	$3 x^{2}$	+	$19 x$
					-	$3 x^{2}$	-	$x$
							+	$18 x$	+	$6$
							-	$18 x$	-	$6$
										$0$

Step 17

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

$2 x^{2} + x + 6$

Divide Using Long Polynomial Division Use the long division method to find the result when 6x^3+5x^2+19x+6 is divided by 3x+1

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