Algebra Examples

$9 x^{3} - 7 x + 2$ , $x - \frac{1}{3}$

Step 1

Divide the higher order polynomial by the other polynomial in order to find the remainder.

$\frac{9 x^{3} - 7 x + 2}{x - \frac{1}{3}}$

Step 2

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

$x$

$\frac{1}{3}$

$9 x^{3}$

$0 x^{2}$

$7 x$

$2$

Step 3

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

					$9 x^{2}$
	$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$

Step 4

Multiply the new quotient term by the divisor.

				$9 x^{2}$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			+	$9 x^{3}$	-	$3 x^{2}$

Step 5

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

				$9 x^{2}$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$

Step 6

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

				$9 x^{2}$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$

Step 7

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

				$9 x^{2}$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$

Step 8

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

				$9 x^{2}$	+	$3 x$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$

Step 9

Multiply the new quotient term by the divisor.

				$9 x^{2}$	+	$3 x$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$x$

Step 10

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

				$9 x^{2}$	+	$3 x$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$x$

Step 11

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

				$9 x^{2}$	+	$3 x$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$x$
							-	$6 x$

Step 12

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

				$9 x^{2}$	+	$3 x$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$x$
							-	$6 x$	+	$2$

Step 13

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

				$9 x^{2}$	+	$3 x$	-	$6$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$x$
							-	$6 x$	+	$2$

Step 14

Multiply the new quotient term by the divisor.

				$9 x^{2}$	+	$3 x$	-	$6$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$x$
							-	$6 x$	+	$2$
							-	$6 x$	+	$2$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 2$

				$9 x^{2}$	+	$3 x$	-	$6$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$x$
							-	$6 x$	+	$2$
							+	$6 x$	-	$2$

Step 16

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

				$9 x^{2}$	+	$3 x$	-	$6$
$x$	-	$\frac{1}{3}$		$9 x^{3}$	+	$0 x^{2}$	-	$7 x$	+	$2$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$x$
							-	$6 x$	+	$2$
							+	$6 x$	-	$2$
										$0$

Step 17

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

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

Step 18

The remainder is the part of the answer that is left after the division by $x - \frac{1}{3}$ is complete.

$0$