Algebra Examples

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

Step 1

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

$2 x$

$1$

$4 x^{3}$

$4 x^{2}$

$3 x$

$0$

Step 2

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

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

Step 3

Multiply the new quotient term by the divisor.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply the new quotient term by the divisor.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $2 x$ .

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

Step 13

Multiply the new quotient term by the divisor.

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

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 2$

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

Step 15

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

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

Step 16

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

$2 x^{2} + x - 2 + \frac{2}{2 x + 1}$