Precalculus Examples

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

Step 1

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

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

Step 2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x^{2}$ .

$3 x^{2}$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{2}$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + 0 - 9 x^{2}$

$3 x^{2}$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

Step 5

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

$3 x^{2}$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

Step 6

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

$3 x^{2}$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x^{2}$ .

$3 x^{2}$

$4 x$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{2}$

$4 x$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

$0 x$

$4 x^{3}$

$0$

$12 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} + 0 + 12 x$

$3 x^{2}$

$4 x$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

$0 x$

$4 x^{3}$

$0$

$12 x$

Step 10

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

$3 x^{2}$

$4 x$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

$0 x$

$4 x^{3}$

$0$

$12 x$

$11 x^{2}$

$12 x$

Step 11

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

$3 x^{2}$

$4 x$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

$0 x$

$4 x^{3}$

$0$

$12 x$

$11 x^{2}$

$12 x$

$3$

Step 12

Divide the highest order term in the dividend $11 x^{2}$ by the highest order term in divisor $x^{2}$ .

$3 x^{2}$

$4 x$

$11$

$x^{2}$

$0 x$

$3$

$3 x^{4}$

$4 x^{3}$

$2 x^{2}$

$0 x$

$3$

$3 x^{4}$

$0$

$9 x^{2}$

$4 x^{3}$

$11 x^{2}$

$0 x$

$4 x^{3}$

$0$

$12 x$

$11 x^{2}$

$12 x$

$3$

Step 13

Multiply the new quotient term by the divisor.

						$3 x^{2}$	-	$4 x$	+	$11$
$x^{2}$	+	$0 x$	-	$3$		$3 x^{4}$	-	$4 x^{3}$	+	$2 x^{2}$	+	$0 x$	-	$3$
					-	$3 x^{4}$	-	$0$	+	$9 x^{2}$
							-	$4 x^{3}$	+	$11 x^{2}$	+	$0 x$
							+	$4 x^{3}$	-	$0$	-	$12 x$
									+	$11 x^{2}$	-	$12 x$	-	$3$
									+	$11 x^{2}$	+	$0$	-	$33$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $11 x^{2} + 0 - 33$

						$3 x^{2}$	-	$4 x$	+	$11$
$x^{2}$	+	$0 x$	-	$3$		$3 x^{4}$	-	$4 x^{3}$	+	$2 x^{2}$	+	$0 x$	-	$3$
					-	$3 x^{4}$	-	$0$	+	$9 x^{2}$
							-	$4 x^{3}$	+	$11 x^{2}$	+	$0 x$
							+	$4 x^{3}$	-	$0$	-	$12 x$
									+	$11 x^{2}$	-	$12 x$	-	$3$
									-	$11 x^{2}$	-	$0$	+	$33$

Step 15

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

						$3 x^{2}$	-	$4 x$	+	$11$
$x^{2}$	+	$0 x$	-	$3$		$3 x^{4}$	-	$4 x^{3}$	+	$2 x^{2}$	+	$0 x$	-	$3$
					-	$3 x^{4}$	-	$0$	+	$9 x^{2}$
							-	$4 x^{3}$	+	$11 x^{2}$	+	$0 x$
							+	$4 x^{3}$	-	$0$	-	$12 x$
									+	$11 x^{2}$	-	$12 x$	-	$3$
									-	$11 x^{2}$	-	$0$	+	$33$
											-	$12 x$	+	$30$

Step 16

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

$3 x^{2} - 4 x + 11 + \frac{- 12 x + 30}{x^{2} - 3}$