Precalculus Examples

$\frac{3 x^{3} + 9 x^{2} - 5 x - 1}{x + 4}$

Step 1

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

$x$

$4$

$3 x^{3}$

$9 x^{2}$

$5 x$

$1$

Step 2

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

					$3 x^{2}$
	$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$

Step 3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			+	$3 x^{3}$	+	$12 x^{2}$

Step 4

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

				$3 x^{2}$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$

Step 5

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

				$3 x^{2}$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$

Step 6

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

				$3 x^{2}$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$

Step 7

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

				$3 x^{2}$	-	$3 x$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$

Step 8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$3 x$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					-	$3 x^{2}$	-	$12 x$

Step 9

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

				$3 x^{2}$	-	$3 x$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$12 x$

Step 10

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

				$3 x^{2}$	-	$3 x$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$12 x$
							+	$7 x$

Step 11

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

				$3 x^{2}$	-	$3 x$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$12 x$
							+	$7 x$	-	$1$

Step 12

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

				$3 x^{2}$	-	$3 x$	+	$7$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$12 x$
							+	$7 x$	-	$1$

Step 13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$3 x$	+	$7$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$12 x$
							+	$7 x$	-	$1$
							+	$7 x$	+	$28$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $7 x + 28$

				$3 x^{2}$	-	$3 x$	+	$7$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$12 x$
							+	$7 x$	-	$1$
							-	$7 x$	-	$28$

Step 15

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

				$3 x^{2}$	-	$3 x$	+	$7$
$x$	+	$4$		$3 x^{3}$	+	$9 x^{2}$	-	$5 x$	-	$1$
			-	$3 x^{3}$	-	$12 x^{2}$
					-	$3 x^{2}$	-	$5 x$
					+	$3 x^{2}$	+	$12 x$
							+	$7 x$	-	$1$
							-	$7 x$	-	$28$
									-	$29$

Step 16

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

$3 x^{2} - 3 x + 7 - \frac{29}{x + 4}$