Precalculus Examples

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

Step 1

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

$3 x$

$6$

$x^{3}$

$3 x^{2}$

$4 x$

$3$

Step 2

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

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

Step 3

Multiply the new quotient term by the divisor.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply the new quotient term by the divisor.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Multiply the new quotient term by the divisor.

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

Step 14

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

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

Step 15

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

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

Step 16

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

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