Algebra Examples

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

Step 1

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

$x^{2}$

$0 x$

$1$

$x^{3}$

$3 x^{2}$

$x$

$3$

Step 2

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$3 x^{2}$

$x$

$3$

Step 3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{3}$

$0$

$x$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply the new quotient term by the divisor.

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

Step 9

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

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

Step 10

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

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

Step 11

Since the remander is $0$ , the final answer is the quotient.

$x - 3$