Algebra Examples

$\frac{(a^{4} - a^{2} - 2 a - 1)}{a^{2} + a + 1}$

Step 1

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

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

Step 2

Divide the highest order term in the dividend $a^{4}$ by the highest order term in divisor $a^{2}$ .

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

Step 3

Multiply the new quotient term by the divisor.

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

$a^{4}$

$a^{3}$

$a^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $a^{4} + a^{3} + a^{2}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

$a^{4}$

$a^{3}$

$a^{2}$

Step 5

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

						$a^{2}$
$a^{2}$	+	$a$	+	$1$		$a^{4}$	+	$0 a^{3}$	-	$a^{2}$	-	$2 a$	-	$1$
					-	$a^{4}$	-	$a^{3}$	-	$a^{2}$
							-	$a^{3}$	-	$2 a^{2}$

Step 6

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

						$a^{2}$
$a^{2}$	+	$a$	+	$1$		$a^{4}$	+	$0 a^{3}$	-	$a^{2}$	-	$2 a$	-	$1$
					-	$a^{4}$	-	$a^{3}$	-	$a^{2}$
							-	$a^{3}$	-	$2 a^{2}$	-	$2 a$

Step 7

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

						$a^{2}$	-	$a$
$a^{2}$	+	$a$	+	$1$		$a^{4}$	+	$0 a^{3}$	-	$a^{2}$	-	$2 a$	-	$1$
					-	$a^{4}$	-	$a^{3}$	-	$a^{2}$
							-	$a^{3}$	-	$2 a^{2}$	-	$2 a$

Step 8

Multiply the new quotient term by the divisor.

						$a^{2}$	-	$a$
$a^{2}$	+	$a$	+	$1$		$a^{4}$	+	$0 a^{3}$	-	$a^{2}$	-	$2 a$	-	$1$
					-	$a^{4}$	-	$a^{3}$	-	$a^{2}$
							-	$a^{3}$	-	$2 a^{2}$	-	$2 a$
							-	$a^{3}$	-	$a^{2}$	-	$a$

Step 9

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

$a^{2}$

$a$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

$a^{4}$

$a^{3}$

$a^{2}$

$a^{3}$

$2 a^{2}$

$2 a$

$a^{3}$

$a^{2}$

$a$

Step 10

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

$a^{2}$

$a$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

$a^{4}$

$a^{3}$

$a^{2}$

$a^{3}$

$2 a^{2}$

$2 a$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

Step 11

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

$a^{2}$

$a$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

$a^{4}$

$a^{3}$

$a^{2}$

$a^{3}$

$2 a^{2}$

$2 a$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$1$

Step 12

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

						$a^{2}$	-	$a$	-	$1$
$a^{2}$	+	$a$	+	$1$		$a^{4}$	+	$0 a^{3}$	-	$a^{2}$	-	$2 a$	-	$1$
					-	$a^{4}$	-	$a^{3}$	-	$a^{2}$
							-	$a^{3}$	-	$2 a^{2}$	-	$2 a$
							+	$a^{3}$	+	$a^{2}$	+	$a$
									-	$a^{2}$	-	$a$	-	$1$

Step 13

Multiply the new quotient term by the divisor.

						$a^{2}$	-	$a$	-	$1$
$a^{2}$	+	$a$	+	$1$		$a^{4}$	+	$0 a^{3}$	-	$a^{2}$	-	$2 a$	-	$1$
					-	$a^{4}$	-	$a^{3}$	-	$a^{2}$
							-	$a^{3}$	-	$2 a^{2}$	-	$2 a$
							+	$a^{3}$	+	$a^{2}$	+	$a$
									-	$a^{2}$	-	$a$	-	$1$
									-	$a^{2}$	-	$a$	-	$1$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- a^{2} - a - 1$

$a^{2}$

$a$

$1$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$a^{2}$

$2 a$

$1$

$a^{4}$

$a^{3}$

$a^{2}$

$a^{3}$

$2 a^{2}$

$2 a$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$1$

$a^{2}$

$a$

$1$

Step 15

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

						$a^{2}$	-	$a$	-	$1$
$a^{2}$	+	$a$	+	$1$		$a^{4}$	+	$0 a^{3}$	-	$a^{2}$	-	$2 a$	-	$1$
					-	$a^{4}$	-	$a^{3}$	-	$a^{2}$
							-	$a^{3}$	-	$2 a^{2}$	-	$2 a$
							+	$a^{3}$	+	$a^{2}$	+	$a$
									-	$a^{2}$	-	$a$	-	$1$
									+	$a^{2}$	+	$a$	+	$1$
														$0$

Step 16

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

$a^{2} - a - 1$