Algebra Examples

$(- 5 x^{3} + 4 x^{2} - 8) \div (x^{2} - 2 x + 2)$

Step 1

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

$x^{2}$

$2 x$

$2$

$5 x^{3}$

$4 x^{2}$

$0 x$

$8$

Step 2

Divide the highest order term in the dividend $- 5 x^{3}$ by the highest order term in divisor $x^{2}$ .

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

Step 3

Multiply the new quotient term by the divisor.

$5 x$

$x^{2}$

$2 x$

$2$

$5 x^{3}$

$4 x^{2}$

$0 x$

$8$

$5 x^{3}$

$10 x^{2}$

$10 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{3} + 10 x^{2} - 10 x$

$5 x$

$x^{2}$

$2 x$

$2$

$5 x^{3}$

$4 x^{2}$

$0 x$

$8$

$5 x^{3}$

$10 x^{2}$

$10 x$

Step 5

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

$5 x$

$x^{2}$

$2 x$

$2$

$5 x^{3}$

$4 x^{2}$

$0 x$

$8$

$5 x^{3}$

$10 x^{2}$

$10 x$

$6 x^{2}$

$10 x$

Step 6

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

$5 x$

$x^{2}$

$2 x$

$2$

$5 x^{3}$

$4 x^{2}$

$0 x$

$8$

$5 x^{3}$

$10 x^{2}$

$10 x$

$6 x^{2}$

$10 x$

$8$

Step 7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x^{2}$ .

					-	$5 x$	-	$6$
$x^{2}$	-	$2 x$	+	$2$	-	$5 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$8$
					+	$5 x^{3}$	-	$10 x^{2}$	+	$10 x$
							-	$6 x^{2}$	+	$10 x$	-	$8$

Step 8

Multiply the new quotient term by the divisor.

					-	$5 x$	-	$6$
$x^{2}$	-	$2 x$	+	$2$	-	$5 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$8$
					+	$5 x^{3}$	-	$10 x^{2}$	+	$10 x$
							-	$6 x^{2}$	+	$10 x$	-	$8$
							-	$6 x^{2}$	+	$12 x$	-	$12$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 12 x - 12$

					-	$5 x$	-	$6$
$x^{2}$	-	$2 x$	+	$2$	-	$5 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$8$
					+	$5 x^{3}$	-	$10 x^{2}$	+	$10 x$
							-	$6 x^{2}$	+	$10 x$	-	$8$
							+	$6 x^{2}$	-	$12 x$	+	$12$

Step 10

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

					-	$5 x$	-	$6$
$x^{2}$	-	$2 x$	+	$2$	-	$5 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$8$
					+	$5 x^{3}$	-	$10 x^{2}$	+	$10 x$
							-	$6 x^{2}$	+	$10 x$	-	$8$
							+	$6 x^{2}$	-	$12 x$	+	$12$
									-	$2 x$	+	$4$

Step 11

The quotient is $- 5 x - 6$ .

$- 5 x - 6$