Algebra Examples

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

Step 1

Move $- 2 x$ .

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

Step 2

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

$x$

$2$

$2 x^{3}$

$9 x^{2}$

$2 x$

$k$

Step 3

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

					$2 x^{2}$
	$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$

Step 4

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			+	$2 x^{3}$	-	$4 x^{2}$

Step 5

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$

Step 6

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$

Step 7

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$

Step 8

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$5 x$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$

Step 9

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					-	$5 x^{2}$	+	$10 x$

Step 10

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

				$2 x^{2}$	-	$5 x$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					+	$5 x^{2}$	-	$10 x$

Step 11

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

				$2 x^{2}$	-	$5 x$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					+	$5 x^{2}$	-	$10 x$
							-	$12 x$

Step 12

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

				$2 x^{2}$	-	$5 x$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					+	$5 x^{2}$	-	$10 x$
							-	$12 x$	+	$k$

Step 13

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					+	$5 x^{2}$	-	$10 x$
							-	$12 x$	+	$k$

Step 14

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					+	$5 x^{2}$	-	$10 x$
							-	$12 x$	+	$k$
							-	$12 x$	+	$24$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x + 24$

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					+	$5 x^{2}$	-	$10 x$
							-	$12 x$	+	$k$
							+	$12 x$	-	$24$

Step 16

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

				$2 x^{2}$	-	$5 x$	-	$12$
$x$	-	$2$		$2 x^{3}$	-	$9 x^{2}$	-	$2 x$	+	$k$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	-	$2 x$
					+	$5 x^{2}$	-	$10 x$
							-	$12 x$	+	$k$
							+	$12 x$	-	$24$
									+	$k$	-	$24$

Step 17

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

$2 x^{2} - 5 x - 12 + \frac{k - 24}{x - 2}$