Algebra Examples

4 x^{3} + 11 x^{2} - 8 x - 10

$4 x^{3} + 11 x^{2} - 8 x - 10$ ,

x + 3

$x + 3$

Step 1

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

\frac{4 x^{3} + 11 x^{2} - 8 x - 10}{x + 3}

$\frac{4 x^{3} + 11 x^{2} - 8 x - 10}{x + 3}$

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

$0$ .

x

$x$

3

$3$

4 x^{3}

$4 x^{3}$

11 x^{2}

$11 x^{2}$

8 x

$8 x$

10

$10$

Step 3

Divide the highest order term in the dividend

4 x^{3}

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

x

$x$ .

					$4 x^{2}$ $4 x^{2}$
	$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$

Step 4

Multiply the new quotient term by the divisor.

				$4 x^{2}$ $4 x^{2}$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			+	$4 x^{3}$ $4 x^{3}$	+	$12 x^{2}$ $12 x^{2}$

Step 5

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

4 x^{3} + 12 x^{2}

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

				$4 x^{2}$ $4 x^{2}$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$

Step 6

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

				$4 x^{2}$ $4 x^{2}$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$

Step 7

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

				$4 x^{2}$ $4 x^{2}$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$

Step 8

Divide the highest order term in the dividend

- x^{2}

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

x

$x$ .

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$

Step 9

Multiply the new quotient term by the divisor.

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
					-	$x^{2}$ $x^{2}$	-	$3 x$ $3 x$

Step 10

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

- x^{2} - 3 x

$- x^{2} - 3 x$

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
					+	$x^{2}$ $x^{2}$	+	$3 x$ $3 x$

Step 11

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

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
					+	$x^{2}$ $x^{2}$	+	$3 x$ $3 x$
							-	$5 x$ $5 x$

Step 12

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

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
					+	$x^{2}$ $x^{2}$	+	$3 x$ $3 x$
							-	$5 x$ $5 x$	-	$10$ $10$

Step 13

Divide the highest order term in the dividend

- 5 x

$- 5 x$ by the highest order term in divisor

x

$x$ .

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$	-	$5$ $5$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
					+	$x^{2}$ $x^{2}$	+	$3 x$ $3 x$
							-	$5 x$ $5 x$	-	$10$ $10$

Step 14

Multiply the new quotient term by the divisor.

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$	-	$5$ $5$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
					+	$x^{2}$ $x^{2}$	+	$3 x$ $3 x$
							-	$5 x$ $5 x$	-	$10$ $10$
							-	$5 x$ $5 x$	-	$15$ $15$

Step 15

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

- 5 x - 15

$- 5 x - 15$

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$	-	$5$ $5$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$ $11 x^{2}$	-	$8 x$ $8 x$	-	$10$ $10$
			-	$4 x^{3}$ $4 x^{3}$	-	$12 x^{2}$ $12 x^{2}$
					-	$x^{2}$ $x^{2}$	-	$8 x$ $8 x$
					+	$x^{2}$ $x^{2}$	+	$3 x$ $3 x$
							-	$5 x$ $5 x$	-	$10$ $10$
							+	$5 x$ $5 x$	+	$15$ $15$

Step 16

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

				$4 x^{2}$ $4 x^{2}$	-	$x$ $x$	-	$5$ $5$
$x$ $x$	+	$3$ $3$		$4 x^{3}$ $4 x^{3}$	+	$11 x^{2}$	-	$8 x$	-	$10$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$x^{2}$	-	$8 x$
					+	$x^{2}$	+	$3 x$
							-	$5 x$	-	$10$
							+	$5 x$	+	$15$
									+	$5$

Step 17

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

$4 x^{2} - x - 5 + \frac{5}{x + 3}$

Step 18

The remainder is the part of the answer that is left after the division by

$x + 3$ is complete.

$5$

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