Algebra Examples

$\frac{2 x^{3} + 11 x^{2} - 21 x}{x^{2} + 3 x}$

Step 1

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

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

Step 2

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

$2 x$

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$2 x$

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

$2 x^{3}$

$6 x^{2}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 6 x^{2} + 0$

$2 x$

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

$2 x^{3}$

$6 x^{2}$

$0$

Step 5

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

$2 x$

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

$2 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$21 x$

Step 6

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

$2 x$

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

$2 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$21 x$

$0$

Step 7

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

$2 x$

$5$

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

$2 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$21 x$

$0$

Step 8

Multiply the new quotient term by the divisor.

$2 x$

$5$

$x^{2}$

$3 x$

$0$

$2 x^{3}$

$11 x^{2}$

$21 x$

$0$

$2 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$21 x$

$0$

$5 x^{2}$

$15 x$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 15 x + 0$

						$2 x$	+	$5$
$x^{2}$	+	$3 x$	+	$0$		$2 x^{3}$	+	$11 x^{2}$	-	$21 x$	+	$0$
					-	$2 x^{3}$	-	$6 x^{2}$	-	$0$
							+	$5 x^{2}$	-	$21 x$	+	$0$
							-	$5 x^{2}$	-	$15 x$	-	$0$

Step 10

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

						$2 x$	+	$5$
$x^{2}$	+	$3 x$	+	$0$		$2 x^{3}$	+	$11 x^{2}$	-	$21 x$	+	$0$
					-	$2 x^{3}$	-	$6 x^{2}$	-	$0$
							+	$5 x^{2}$	-	$21 x$	+	$0$
							-	$5 x^{2}$	-	$15 x$	-	$0$
									-	$36 x$	+	$0$

Step 11

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

$2 x + 5 + \frac{- 36 x}{x^{2} + 3 x}$