Algebra Examples

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

Step 1

Reorder $3 x^{2}$ and $- 2 x^{3}$ .

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

Step 2

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

$x$

$2$

$2 x^{3}$

$3 x^{2}$

$0 x$

$1$

Step 3

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

				-	$2 x^{2}$
	$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$

Step 4

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			-	$2 x^{3}$	-	$4 x^{2}$

Step 5

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

			-	$2 x^{2}$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$

Step 6

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

			-	$2 x^{2}$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$

Step 7

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

			-	$2 x^{2}$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$

Step 8

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

			-	$2 x^{2}$	+	$7 x$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$

Step 9

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$	+	$7 x$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					+	$7 x^{2}$	+	$14 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 14 x$

			-	$2 x^{2}$	+	$7 x$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	-	$14 x$

Step 11

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

			-	$2 x^{2}$	+	$7 x$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	-	$14 x$
							-	$14 x$

Step 12

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

			-	$2 x^{2}$	+	$7 x$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	-	$14 x$
							-	$14 x$	-	$1$

Step 13

Divide the highest order term in the dividend $- 14 x$ by the highest order term in divisor $x$ .

			-	$2 x^{2}$	+	$7 x$	-	$14$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	-	$14 x$
							-	$14 x$	-	$1$

Step 14

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$	+	$7 x$	-	$14$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	-	$14 x$
							-	$14 x$	-	$1$
							-	$14 x$	-	$28$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x - 28$

			-	$2 x^{2}$	+	$7 x$	-	$14$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	-	$14 x$
							-	$14 x$	-	$1$
							+	$14 x$	+	$28$

Step 16

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

			-	$2 x^{2}$	+	$7 x$	-	$14$
$x$	+	$2$	-	$2 x^{3}$	+	$3 x^{2}$	+	$0 x$	-	$1$
			+	$2 x^{3}$	+	$4 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	-	$14 x$
							-	$14 x$	-	$1$
							+	$14 x$	+	$28$
									+	$27$

Step 17

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

$- 2 x^{2} + 7 x - 14 + \frac{27}{x + 2}$