Precalculus Examples

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

Step 1

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

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

Step 2

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + 15 x^{3}$

				$3 x^{3}$
$x$	+	$5$		$3 x^{4}$	-	$x^{3}$	-	$4 x^{2}$	+	$9 x$	-	$3$
			-	$3 x^{4}$	-	$15 x^{3}$

Step 5

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

				$3 x^{3}$
$x$	+	$5$		$3 x^{4}$	-	$x^{3}$	-	$4 x^{2}$	+	$9 x$	-	$3$
			-	$3 x^{4}$	-	$15 x^{3}$
					-	$16 x^{3}$

Step 6

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

				$3 x^{3}$
$x$	+	$5$		$3 x^{4}$	-	$x^{3}$	-	$4 x^{2}$	+	$9 x$	-	$3$
			-	$3 x^{4}$	-	$15 x^{3}$
					-	$16 x^{3}$	-	$4 x^{2}$

Step 7

Divide the highest order term in the dividend $- 16 x^{3}$ by the highest order term in divisor $x$ .

				$3 x^{3}$	-	$16 x^{2}$
$x$	+	$5$		$3 x^{4}$	-	$x^{3}$	-	$4 x^{2}$	+	$9 x$	-	$3$
			-	$3 x^{4}$	-	$15 x^{3}$
					-	$16 x^{3}$	-	$4 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

				$3 x^{3}$	-	$16 x^{2}$
$x$	+	$5$		$3 x^{4}$	-	$x^{3}$	-	$4 x^{2}$	+	$9 x$	-	$3$
			-	$3 x^{4}$	-	$15 x^{3}$
					-	$16 x^{3}$	-	$4 x^{2}$
					-	$16 x^{3}$	-	$80 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x^{3} - 80 x^{2}$

$3 x^{3}$

$16 x^{2}$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

Step 10

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

$3 x^{3}$

$16 x^{2}$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

Step 11

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

$3 x^{3}$

$16 x^{2}$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

Step 12

Divide the highest order term in the dividend $76 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$16 x^{2}$

$76 x$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$16 x^{2}$

$76 x$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

$76 x^{2}$

$380 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $76 x^{2} + 380 x$

$3 x^{3}$

$16 x^{2}$

$76 x$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

$76 x^{2}$

$380 x$

Step 15

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

$3 x^{3}$

$16 x^{2}$

$76 x$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

$76 x^{2}$

$380 x$

$371 x$

Step 16

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

$3 x^{3}$

$16 x^{2}$

$76 x$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

$76 x^{2}$

$380 x$

$371 x$

$3$

Step 17

Divide the highest order term in the dividend $- 371 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$16 x^{2}$

$76 x$

$371$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

$76 x^{2}$

$380 x$

$371 x$

$3$

Step 18

Multiply the new quotient term by the divisor.

				$3 x^{3}$	-	$16 x^{2}$	+	$76 x$	-	$371$
$x$	+	$5$		$3 x^{4}$	-	$x^{3}$	-	$4 x^{2}$	+	$9 x$	-	$3$
			-	$3 x^{4}$	-	$15 x^{3}$
					-	$16 x^{3}$	-	$4 x^{2}$
					+	$16 x^{3}$	+	$80 x^{2}$
							+	$76 x^{2}$	+	$9 x$
							-	$76 x^{2}$	-	$380 x$
									-	$371 x$	-	$3$
									-	$371 x$	-	$1855$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 371 x - 1855$

$3 x^{3}$

$16 x^{2}$

$76 x$

$371$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

$76 x^{2}$

$380 x$

$371 x$

$3$

$371 x$

$1855$

Step 20

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

$3 x^{3}$

$16 x^{2}$

$76 x$

$371$

$x$

$5$

$3 x^{4}$

$x^{3}$

$4 x^{2}$

$9 x$

$3$

$3 x^{4}$

$15 x^{3}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$80 x^{2}$

$76 x^{2}$

$9 x$

$76 x^{2}$

$380 x$

$371 x$

$3$

$371 x$

$1855$

$1852$

Step 21

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

$3 x^{3} - 16 x^{2} + 76 x - 371 + \frac{1852}{x + 5}$