Pre-Algebra Examples

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

Step 1

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

$x$

$2$

$4 x^{3}$

$5 x^{2}$

$3 x$

$2$

Step 2

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

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

Step 3

Multiply the new quotient term by the divisor.

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

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 8 x^{2}$

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

Step 5

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

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

Step 6

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

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

Step 7

Divide the highest order term in the dividend $- 13 x^{2}$ by the highest order term in divisor $x$ .

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

Step 8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$13 x$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					-	$13 x^{2}$	-	$26 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{2} - 26 x$

				$4 x^{2}$	-	$13 x$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					+	$13 x^{2}$	+	$26 x$

Step 10

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

				$4 x^{2}$	-	$13 x$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					+	$13 x^{2}$	+	$26 x$
							+	$29 x$

Step 11

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

				$4 x^{2}$	-	$13 x$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					+	$13 x^{2}$	+	$26 x$
							+	$29 x$	-	$2$

Step 12

Divide the highest order term in the dividend $29 x$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$13 x$	+	$29$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					+	$13 x^{2}$	+	$26 x$
							+	$29 x$	-	$2$

Step 13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$13 x$	+	$29$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					+	$13 x^{2}$	+	$26 x$
							+	$29 x$	-	$2$
							+	$29 x$	+	$58$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $29 x + 58$

				$4 x^{2}$	-	$13 x$	+	$29$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					+	$13 x^{2}$	+	$26 x$
							+	$29 x$	-	$2$
							-	$29 x$	-	$58$

Step 15

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

				$4 x^{2}$	-	$13 x$	+	$29$
$x$	+	$2$		$4 x^{3}$	-	$5 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	-	$8 x^{2}$
					-	$13 x^{2}$	+	$3 x$
					+	$13 x^{2}$	+	$26 x$
							+	$29 x$	-	$2$
							-	$29 x$	-	$58$
									-	$60$

Step 16

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

$4 x^{2} - 13 x + 29 - \frac{60}{x + 2}$