Algebra Examples

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

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

Move $2 x$ .

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

Step 1.2

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

$x$

$1$

$x^{3}$

$2 x^{2}$

$2 x$

$k$

Step 1.3

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$

Step 1.4

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			+	$x^{3}$	-	$x^{2}$

Step 1.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$

Step 1.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$

Step 1.7

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$

Step 1.8

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$

Step 1.9

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$
					-	$x^{2}$	+	$x$

Step 1.10

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$x$

Step 1.11

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$x$
							+	$x$

Step 1.12

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$x$
							+	$x$	+	$k$

Step 1.13

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

				$x^{2}$	-	$x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$x$
							+	$x$	+	$k$

Step 1.14

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$x$
							+	$x$	+	$k$
							+	$x$	-	$1$

Step 1.15

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

				$x^{2}$	-	$x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	+	$2 x$	+	$k$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$x$
							+	$x$	+	$k$
							-	$x$	+	$1$

Step 1.16

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

$x^{2}$

$x$

$1$

$x$

$1$

$x^{3}$

$2 x^{2}$

$2 x$

$k$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

$k$

$x$

$1$

$k$

$1$

Step 1.17

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

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

Step 2

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

$k + 1$