Algebra Examples

$\frac{- 19 n^{3} - 22 n^{2} - 18 n - 15}{n + 1}$

Step 1

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

$n$

$1$

$19 n^{3}$

$22 n^{2}$

$18 n$

$15$

Step 2

Divide the highest order term in the dividend $- 19 n^{3}$ by the highest order term in divisor $n$ .

				-	$19 n^{2}$
	$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$

Step 3

Multiply the new quotient term by the divisor.

			-	$19 n^{2}$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			-	$19 n^{3}$	-	$19 n^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 19 n^{3} - 19 n^{2}$

			-	$19 n^{2}$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$

Step 5

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

			-	$19 n^{2}$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$

Step 6

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

			-	$19 n^{2}$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$

Step 7

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

			-	$19 n^{2}$	-	$3 n$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$

Step 8

Multiply the new quotient term by the divisor.

			-	$19 n^{2}$	-	$3 n$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					-	$3 n^{2}$	-	$3 n$

Step 9

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

			-	$19 n^{2}$	-	$3 n$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					+	$3 n^{2}$	+	$3 n$

Step 10

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

			-	$19 n^{2}$	-	$3 n$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					+	$3 n^{2}$	+	$3 n$
							-	$15 n$

Step 11

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

			-	$19 n^{2}$	-	$3 n$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					+	$3 n^{2}$	+	$3 n$
							-	$15 n$	-	$15$

Step 12

Divide the highest order term in the dividend $- 15 n$ by the highest order term in divisor $n$ .

			-	$19 n^{2}$	-	$3 n$	-	$15$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					+	$3 n^{2}$	+	$3 n$
							-	$15 n$	-	$15$

Step 13

Multiply the new quotient term by the divisor.

			-	$19 n^{2}$	-	$3 n$	-	$15$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					+	$3 n^{2}$	+	$3 n$
							-	$15 n$	-	$15$
							-	$15 n$	-	$15$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 n - 15$

			-	$19 n^{2}$	-	$3 n$	-	$15$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					+	$3 n^{2}$	+	$3 n$
							-	$15 n$	-	$15$
							+	$15 n$	+	$15$

Step 15

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

			-	$19 n^{2}$	-	$3 n$	-	$15$
$n$	+	$1$	-	$19 n^{3}$	-	$22 n^{2}$	-	$18 n$	-	$15$
			+	$19 n^{3}$	+	$19 n^{2}$
					-	$3 n^{2}$	-	$18 n$
					+	$3 n^{2}$	+	$3 n$
							-	$15 n$	-	$15$
							+	$15 n$	+	$15$
										$0$

Step 16

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

$- 19 n^{2} - 3 n - 15$