Examples

\frac{x^{3} - x^{2} + 7 x}{6 x - 5}

$\frac{x^{3} - x^{2} + 7 x}{6 x - 5}$

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

$0$ .

6 x

$6 x$

5

$5$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

7 x

$7 x$

0

$0$

Step 2

Divide the highest order term in the dividend

x^{3}

$x^{3}$ by the highest order term in divisor

6 x

$6 x$ .

					$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$
	$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$

Step 3

Multiply the new quotient term by the divisor.

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			+	$x^{3}$ $x^{3}$	-	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

x^{3} - \frac{5 x^{2}}{6}

$x^{3} - \frac{5 x^{2}}{6}$

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$

Step 5

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

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$

Step 6

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

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$

Step 7

Divide the highest order term in the dividend

- \frac{x^{2}}{6}

$- \frac{x^{2}}{6}$ by the highest order term in divisor

6 x

$6 x$ .

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$

Step 8

Multiply the new quotient term by the divisor.

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$\frac{5 x}{36}$ $\frac{5 x}{36}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in

- \frac{x^{2}}{6} + \frac{5 x}{36}

$- \frac{x^{2}}{6} + \frac{5 x}{36}$

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					+	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{5 x}{36}$ $\frac{5 x}{36}$

Step 10

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

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					+	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{5 x}{36}$ $\frac{5 x}{36}$
							+	$\frac{247 x}{36}$ $\frac{247 x}{36}$

Step 11

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

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					+	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{5 x}{36}$ $\frac{5 x}{36}$
							+	$\frac{247 x}{36}$ $\frac{247 x}{36}$	+	$0$ $0$

Step 12

Divide the highest order term in the dividend

\frac{247 x}{36}

$\frac{247 x}{36}$ by the highest order term in divisor

6 x

$6 x$ .

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$	+	$\frac{247}{216}$ $\frac{247}{216}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					+	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{5 x}{36}$ $\frac{5 x}{36}$
							+	$\frac{247 x}{36}$ $\frac{247 x}{36}$	+	$0$ $0$

Step 13

Multiply the new quotient term by the divisor.

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$	+	$\frac{247}{216}$ $\frac{247}{216}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					+	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{5 x}{36}$ $\frac{5 x}{36}$
							+	$\frac{247 x}{36}$ $\frac{247 x}{36}$	+	$0$ $0$
							+	$\frac{247 x}{36}$ $\frac{247 x}{36}$	-	$\frac{1235}{216}$ $\frac{1235}{216}$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in

\frac{247 x}{36} - \frac{1235}{216}

$\frac{247 x}{36} - \frac{1235}{216}$

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$	+	$\frac{247}{216}$ $\frac{247}{216}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					+	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{5 x}{36}$ $\frac{5 x}{36}$
							+	$\frac{247 x}{36}$ $\frac{247 x}{36}$	+	$0$ $0$
							-	$\frac{247 x}{36}$ $\frac{247 x}{36}$	+	$\frac{1235}{216}$ $\frac{1235}{216}$

Step 15

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

				$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{x}{36}$ $\frac{x}{36}$	+	$\frac{247}{216}$ $\frac{247}{216}$
$6 x$ $6 x$	-	$5$ $5$		$x^{3}$ $x^{3}$	-	$x^{2}$ $x^{2}$	+	$7 x$ $7 x$	+	$0$ $0$
			-	$x^{3}$ $x^{3}$	+	$\frac{5 x^{2}}{6}$ $\frac{5 x^{2}}{6}$
					-	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	+	$7 x$ $7 x$
					+	$\frac{x^{2}}{6}$ $\frac{x^{2}}{6}$	-	$\frac{5 x}{36}$ $\frac{5 x}{36}$
							+	$\frac{247 x}{36}$ $\frac{247 x}{36}$	+	$0$ $0$
							-	$\frac{247 x}{36}$ $\frac{247 x}{36}$	+	$\frac{1235}{216}$ $\frac{1235}{216}$
									+	$\frac{1235}{216}$ $\frac{1235}{216}$

Step 16

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

\frac{x^{2}}{6} - \frac{x}{36} + \frac{247}{216} + \frac{1235}{216 (6 x - 5)}

$\frac{x^{2}}{6} - \frac{x}{36} + \frac{247}{216} + \frac{1235}{216 (6 x - 5)}$

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