Algebra Examples

$\frac{- 3 x^{3} + 12 x^{2} - 18 x + 30}{x - 3}$

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 $0$ .

$x$

$3$

$3 x^{3}$

$12 x^{2}$

$18 x$

$30$

Step 1.2

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

				-	$3 x^{2}$
	$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$

Step 1.3

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			-	$3 x^{3}$	+	$9 x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{3} + 9 x^{2}$

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$

Step 1.5

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

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$

Step 1.6

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

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$

Step 1.7

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

			-	$3 x^{2}$	+	$3 x$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$

Step 1.8

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	+	$3 x$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					+	$3 x^{2}$	-	$9 x$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - 9 x$

			-	$3 x^{2}$	+	$3 x$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					-	$3 x^{2}$	+	$9 x$

Step 1.10

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

			-	$3 x^{2}$	+	$3 x$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					-	$3 x^{2}$	+	$9 x$
							-	$9 x$

Step 1.11

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

			-	$3 x^{2}$	+	$3 x$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					-	$3 x^{2}$	+	$9 x$
							-	$9 x$	+	$30$

Step 1.12

Divide the highest order term in the dividend $- 9 x$ by the highest order term in divisor $x$ .

			-	$3 x^{2}$	+	$3 x$	-	$9$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					-	$3 x^{2}$	+	$9 x$
							-	$9 x$	+	$30$

Step 1.13

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	+	$3 x$	-	$9$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					-	$3 x^{2}$	+	$9 x$
							-	$9 x$	+	$30$
							-	$9 x$	+	$27$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x + 27$

			-	$3 x^{2}$	+	$3 x$	-	$9$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					-	$3 x^{2}$	+	$9 x$
							-	$9 x$	+	$30$
							+	$9 x$	-	$27$

Step 1.15

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

			-	$3 x^{2}$	+	$3 x$	-	$9$
$x$	-	$3$	-	$3 x^{3}$	+	$12 x^{2}$	-	$18 x$	+	$30$
			+	$3 x^{3}$	-	$9 x^{2}$
					+	$3 x^{2}$	-	$18 x$
					-	$3 x^{2}$	+	$9 x$
							-	$9 x$	+	$30$
							+	$9 x$	-	$27$
									+	$3$

Step 1.16

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

$- 3 x^{2} + 3 x - 9 + \frac{3}{x - 3}$

Step 2

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

$3$