Algebra Examples

$\frac{x^{4} + x^{3} - 3 x^{2} - 10 x + 2}{x^{2} + 3 x + 3}$

Step 1

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

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

Step 2

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

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

Step 4

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

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

Step 5

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

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$2 x^{3}$

$6 x^{2}$

Step 6

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

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$2 x^{3}$

$6 x^{2}$

$10 x$

Step 7

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

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$2 x^{3}$

$6 x^{2}$

$10 x$

Step 8

Multiply the new quotient term by the divisor.

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

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} - 6 x^{2} - 6 x$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$2 x^{3}$

$6 x^{2}$

$10 x$

$2 x^{3}$

$6 x^{2}$

$6 x$

Step 10

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

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$2 x^{3}$

$6 x^{2}$

$10 x$

$2 x^{3}$

$6 x^{2}$

$6 x$

$4 x$

Step 11

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

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$3$

$x^{4}$

$x^{3}$

$3 x^{2}$

$10 x$

$2$

$x^{4}$

$3 x^{3}$

$3 x^{2}$

$2 x^{3}$

$6 x^{2}$

$10 x$

$2 x^{3}$

$6 x^{2}$

$6 x$

$4 x$

$2$

Step 12

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

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