Precalculus Examples

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

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 $0$ .

$x$

$3$

$2 x^{3}$

$7 x^{2}$

$4 x$

$5$

Step 1.2

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

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

Step 1.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			+	$2 x^{3}$	-	$6 x^{2}$

Step 1.4

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$

Step 1.5

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$

Step 1.6

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$

Step 1.7

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

				$2 x^{2}$	-	$x$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$

Step 1.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$x$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					-	$x^{2}$	+	$3 x$

Step 1.9

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

				$2 x^{2}$	-	$x$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$3 x$

Step 1.10

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

				$2 x^{2}$	-	$x$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$3 x$
							+	$x$

Step 1.11

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

				$2 x^{2}$	-	$x$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$3 x$
							+	$x$	-	$5$

Step 1.12

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

				$2 x^{2}$	-	$x$	+	$1$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$3 x$
							+	$x$	-	$5$

Step 1.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$x$	+	$1$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$3 x$
							+	$x$	-	$5$
							+	$x$	-	$3$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $x - 3$

				$2 x^{2}$	-	$x$	+	$1$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$3 x$
							+	$x$	-	$5$
							-	$x$	+	$3$

Step 1.15

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

				$2 x^{2}$	-	$x$	+	$1$
$x$	-	$3$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	-	$5$
			-	$2 x^{3}$	+	$6 x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$3 x$
							+	$x$	-	$5$
							-	$x$	+	$3$
									-	$2$

Step 1.16

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

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

Step 2

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

$- 2$