Precalculus Examples

$\frac{4 x^{3} - 8 x^{2} + 2 x - 1}{2 x + 1}$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

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

$2 x$

$1$

$4 x^{3}$

$8 x^{2}$

$2 x$

$1$

Step 1.2

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

					$2 x^{2}$
	$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$

Step 1.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			+	$4 x^{3}$	+	$2 x^{2}$

Step 1.4

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

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$

Step 1.5

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

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$

Step 1.6

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

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$

Step 1.7

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$

Step 1.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					-	$10 x^{2}$	-	$5 x$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 5 x$

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					+	$10 x^{2}$	+	$5 x$

Step 1.10

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

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					+	$10 x^{2}$	+	$5 x$
							+	$7 x$

Step 1.11

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

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					+	$10 x^{2}$	+	$5 x$
							+	$7 x$	-	$1$

Step 1.12

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$5 x$	+	$\frac{7}{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					+	$10 x^{2}$	+	$5 x$
							+	$7 x$	-	$1$

Step 1.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$	+	$\frac{7}{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					+	$10 x^{2}$	+	$5 x$
							+	$7 x$	-	$1$
							+	$7 x$	+	$\frac{7}{2}$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $7 x + \frac{7}{2}$

				$2 x^{2}$	-	$5 x$	+	$\frac{7}{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					+	$10 x^{2}$	+	$5 x$
							+	$7 x$	-	$1$
							-	$7 x$	-	$\frac{7}{2}$

Step 1.15

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

				$2 x^{2}$	-	$5 x$	+	$\frac{7}{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$2 x$	-	$1$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$2 x$
					+	$10 x^{2}$	+	$5 x$
							+	$7 x$	-	$1$
							-	$7 x$	-	$\frac{7}{2}$
									-	$\frac{9}{2}$

Step 1.16

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

$2 x^{2} - 5 x + \frac{7}{2} - \frac{9}{2 (2 x + 1)}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$- 9$