Pre-Algebra Examples

$(12 x^{3} + 11 x^{2} - 22 x - 1) \div (4 x^{2} + x)$

Step 1

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

$4 x^{2}$

$x$

$0$

$12 x^{3}$

$11 x^{2}$

$22 x$

$1$

Step 2

Divide the highest order term in the dividend $12 x^{3}$ by the highest order term in divisor $4 x^{2}$ .

$3 x$

$4 x^{2}$

$x$

$0$

$12 x^{3}$

$11 x^{2}$

$22 x$

$1$

Step 3

Multiply the new quotient term by the divisor.

$3 x$

$4 x^{2}$

$x$

$0$

$12 x^{3}$

$11 x^{2}$

$22 x$

$1$

$12 x^{3}$

$3 x^{2}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{3} + 3 x^{2} + 0$

						$3 x$
$4 x^{2}$	+	$x$	+	$0$		$12 x^{3}$	+	$11 x^{2}$	-	$22 x$	-	$1$
					-	$12 x^{3}$	-	$3 x^{2}$	-	$0$

Step 5

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

$3 x$

$4 x^{2}$

$x$

$0$

$12 x^{3}$

$11 x^{2}$

$22 x$

$1$

$12 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$22 x$

Step 6

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

						$3 x$
$4 x^{2}$	+	$x$	+	$0$		$12 x^{3}$	+	$11 x^{2}$	-	$22 x$	-	$1$
					-	$12 x^{3}$	-	$3 x^{2}$	-	$0$
							+	$8 x^{2}$	-	$22 x$	-	$1$

Step 7

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $4 x^{2}$ .

						$3 x$	+	$2$
$4 x^{2}$	+	$x$	+	$0$		$12 x^{3}$	+	$11 x^{2}$	-	$22 x$	-	$1$
					-	$12 x^{3}$	-	$3 x^{2}$	-	$0$
							+	$8 x^{2}$	-	$22 x$	-	$1$

Step 8

Multiply the new quotient term by the divisor.

$3 x$

$2$

$4 x^{2}$

$x$

$0$

$12 x^{3}$

$11 x^{2}$

$22 x$

$1$

$12 x^{3}$

$3 x^{2}$

$0$

$8 x^{2}$

$22 x$

$1$

$8 x^{2}$

$2 x$

$0$

Step 9

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

						$3 x$	+	$2$
$4 x^{2}$	+	$x$	+	$0$		$12 x^{3}$	+	$11 x^{2}$	-	$22 x$	-	$1$
					-	$12 x^{3}$	-	$3 x^{2}$	-	$0$
							+	$8 x^{2}$	-	$22 x$	-	$1$
							-	$8 x^{2}$	-	$2 x$	-	$0$

Step 10

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

						$3 x$	+	$2$
$4 x^{2}$	+	$x$	+	$0$		$12 x^{3}$	+	$11 x^{2}$	-	$22 x$	-	$1$
					-	$12 x^{3}$	-	$3 x^{2}$	-	$0$
							+	$8 x^{2}$	-	$22 x$	-	$1$
							-	$8 x^{2}$	-	$2 x$	-	$0$
									-	$24 x$	-	$1$

Step 11

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

$3 x + 2 + \frac{- 24 x - 1}{4 x^{2} + x}$