Algebra Examples

$\frac{3 x^{3} + 9 x^{2} - 6 x}{- 3 x}$

Step 1

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

$3 x$

$0$

$3 x^{3}$

$9 x^{2}$

$6 x$

$0$

Step 2

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

				-	$x^{2}$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$

Step 3

Multiply the new quotient term by the divisor.

				-	$x^{2}$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				+	$3 x^{3}$	+	$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} + 0$

				-	$x^{2}$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$

Step 5

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

				-	$x^{2}$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$

Step 6

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

				-	$x^{2}$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$

Step 7

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $- 3 x$ .

				-	$x^{2}$	-	$3 x$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$

Step 8

Multiply the new quotient term by the divisor.

				-	$x^{2}$	-	$3 x$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						+	$9 x^{2}$	+	$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} + 0$

				-	$x^{2}$	-	$3 x$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						-	$9 x^{2}$	-	$0$

Step 10

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

				-	$x^{2}$	-	$3 x$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						-	$9 x^{2}$	-	$0$
								-	$6 x$

Step 11

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

				-	$x^{2}$	-	$3 x$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						-	$9 x^{2}$	-	$0$
								-	$6 x$	+	$0$

Step 12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $- 3 x$ .

				-	$x^{2}$	-	$3 x$	+	$2$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						-	$9 x^{2}$	-	$0$
								-	$6 x$	+	$0$

Step 13

Multiply the new quotient term by the divisor.

				-	$x^{2}$	-	$3 x$	+	$2$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						-	$9 x^{2}$	-	$0$
								-	$6 x$	+	$0$
								-	$6 x$	+	$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 0$

				-	$x^{2}$	-	$3 x$	+	$2$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						-	$9 x^{2}$	-	$0$
								-	$6 x$	+	$0$
								+	$6 x$	-	$0$

Step 15

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

				-	$x^{2}$	-	$3 x$	+	$2$
-	$3 x$	+	$0$		$3 x^{3}$	+	$9 x^{2}$	-	$6 x$	+	$0$
				-	$3 x^{3}$	-	$0$
						+	$9 x^{2}$	-	$6 x$
						-	$9 x^{2}$	-	$0$
								-	$6 x$	+	$0$
								+	$6 x$	-	$0$
											$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

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