Algebra Examples

\frac{(4 x^{3} + 3 x^{2} - 30 x - 10)}{(x - 3)}

$\frac{(4 x^{3} + 3 x^{2} - 30 x - 10)}{(x - 3)}$

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

$0$ .

$x$

$3$

$4 x^{3}$

$3 x^{2}$

$30 x$

$10$

Step 2

Divide the highest order term in the dividend

$4 x^{3}$ by the highest order term in divisor

$x$ .

					$4 x^{2}$
	$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$

Step 3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			+	$4 x^{3}$	-	$12 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

$4 x^{3} - 12 x^{2}$

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$

Step 5

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

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$

Step 6

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

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$

Step 7

Divide the highest order term in the dividend

$15 x^{2}$ by the highest order term in divisor

$x$ .

				$4 x^{2}$	+	$15 x$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$

Step 8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$15 x$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					+	$15 x^{2}$	-	$45 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in

$15 x^{2} - 45 x$

				$4 x^{2}$	+	$15 x$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					-	$15 x^{2}$	+	$45 x$

Step 10

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

				$4 x^{2}$	+	$15 x$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					-	$15 x^{2}$	+	$45 x$
							+	$15 x$

Step 11

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

				$4 x^{2}$	+	$15 x$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					-	$15 x^{2}$	+	$45 x$
							+	$15 x$	-	$10$

Step 12

Divide the highest order term in the dividend

$15 x$ by the highest order term in divisor

$x$ .

				$4 x^{2}$	+	$15 x$	+	$15$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					-	$15 x^{2}$	+	$45 x$
							+	$15 x$	-	$10$

Step 13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$15 x$	+	$15$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					-	$15 x^{2}$	+	$45 x$
							+	$15 x$	-	$10$
							+	$15 x$	-	$45$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in

$15 x - 45$

				$4 x^{2}$	+	$15 x$	+	$15$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					-	$15 x^{2}$	+	$45 x$
							+	$15 x$	-	$10$
							-	$15 x$	+	$45$

Step 15

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

				$4 x^{2}$	+	$15 x$	+	$15$
$x$	-	$3$		$4 x^{3}$	+	$3 x^{2}$	-	$30 x$	-	$10$
			-	$4 x^{3}$	+	$12 x^{2}$
					+	$15 x^{2}$	-	$30 x$
					-	$15 x^{2}$	+	$45 x$
							+	$15 x$	-	$10$
							-	$15 x$	+	$45$
									+	$35$

Step 16

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

$4 x^{2} + 15 x + 15 + \frac{35}{x - 3}$