Algebra Examples

$\frac{(3 x^{3} - 2 x^{2} - 5)}{(x^{2} - 4)}$

Step 1

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

$x^{2}$

$0 x$

$4$

$3 x^{3}$

$2 x^{2}$

$0 x$

$5$

Step 2

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

$3 x$

$x^{2}$

$0 x$

$4$

$3 x^{3}$

$2 x^{2}$

$0 x$

$5$

Step 3

Multiply the new quotient term by the divisor.

$3 x$

$x^{2}$

$0 x$

$4$

$3 x^{3}$

$2 x^{2}$

$0 x$

$5$

$3 x^{3}$

$0$

$12 x$

Step 4

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

						$3 x$
$x^{2}$	+	$0 x$	-	$4$		$3 x^{3}$	-	$2 x^{2}$	+	$0 x$	-	$5$
					-	$3 x^{3}$	-	$0$	+	$12 x$

Step 5

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

$3 x$

$x^{2}$

$0 x$

$4$

$3 x^{3}$

$2 x^{2}$

$0 x$

$5$

$3 x^{3}$

$0$

$12 x$

$2 x^{2}$

$12 x$

Step 6

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

						$3 x$
$x^{2}$	+	$0 x$	-	$4$		$3 x^{3}$	-	$2 x^{2}$	+	$0 x$	-	$5$
					-	$3 x^{3}$	-	$0$	+	$12 x$
							-	$2 x^{2}$	+	$12 x$	-	$5$

Step 7

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

						$3 x$	-	$2$
$x^{2}$	+	$0 x$	-	$4$		$3 x^{3}$	-	$2 x^{2}$	+	$0 x$	-	$5$
					-	$3 x^{3}$	-	$0$	+	$12 x$
							-	$2 x^{2}$	+	$12 x$	-	$5$

Step 8

Multiply the new quotient term by the divisor.

						$3 x$	-	$2$
$x^{2}$	+	$0 x$	-	$4$		$3 x^{3}$	-	$2 x^{2}$	+	$0 x$	-	$5$
					-	$3 x^{3}$	-	$0$	+	$12 x$
							-	$2 x^{2}$	+	$12 x$	-	$5$
							-	$2 x^{2}$	+	$0$	+	$8$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 0 + 8$

						$3 x$	-	$2$
$x^{2}$	+	$0 x$	-	$4$		$3 x^{3}$	-	$2 x^{2}$	+	$0 x$	-	$5$
					-	$3 x^{3}$	-	$0$	+	$12 x$
							-	$2 x^{2}$	+	$12 x$	-	$5$
							+	$2 x^{2}$	-	$0$	-	$8$

Step 10

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

						$3 x$	-	$2$
$x^{2}$	+	$0 x$	-	$4$		$3 x^{3}$	-	$2 x^{2}$	+	$0 x$	-	$5$
					-	$3 x^{3}$	-	$0$	+	$12 x$
							-	$2 x^{2}$	+	$12 x$	-	$5$
							+	$2 x^{2}$	-	$0$	-	$8$
									+	$12 x$	-	$13$

Step 11

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

$3 x - 2 + \frac{12 x - 13}{x^{2} - 4}$