Precalculus Examples

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

Step 1

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

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$9$

Step 2

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$9$

Step 3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$9$

$x^{3}$

$0$

$x$

Step 4

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$9$

$x^{3}$

$0$

$x$

Step 5

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$9$

$x^{3}$

$0$

$x$

Step 6

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

						$x$
$x^{2}$	+	$0 x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$9$
					-	$x^{3}$	-	$0$	-	$x$
									-	$x$	-	$9$

Step 7

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

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