Algebra Examples

$(6 x^{4} - 5 x^{3} - 6 x - 6) \div (x^{2} - x - 1)$

Step 1

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

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

Step 2

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $x^{2}$ .

							$6 x^{2}$
	$x^{2}$	-	$x$	-	$1$		$6 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	-	$6 x$	-	$6$

Step 3

Multiply the new quotient term by the divisor.

						$6 x^{2}$
$x^{2}$	-	$x$	-	$1$		$6 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	-	$6 x$	-	$6$
					+	$6 x^{4}$	-	$6 x^{3}$	-	$6 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} - 6 x^{3} - 6 x^{2}$

$6 x^{2}$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

Step 5

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

$6 x^{2}$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

Step 6

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

$6 x^{2}$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

Step 7

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

$6 x^{2}$

$x$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

Step 8

Multiply the new quotient term by the divisor.

$6 x^{2}$

$x$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

Step 9

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

$6 x^{2}$

$x$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

Step 10

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

$6 x^{2}$

$x$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

$7 x^{2}$

$5 x$

Step 11

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

$6 x^{2}$

$x$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

$7 x^{2}$

$5 x$

$6$

Step 12

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x^{2}$ .

$6 x^{2}$

$x$

$7$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

$7 x^{2}$

$5 x$

$6$

Step 13

Multiply the new quotient term by the divisor.

$6 x^{2}$

$x$

$7$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

$7 x^{2}$

$5 x$

$6$

$7 x^{2}$

$7 x$

$7$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} - 7 x - 7$

$6 x^{2}$

$x$

$7$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

$7 x^{2}$

$5 x$

$6$

$7 x^{2}$

$7 x$

$7$

Step 15

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

$6 x^{2}$

$x$

$7$

$x^{2}$

$x$

$1$

$6 x^{4}$

$5 x^{3}$

$0 x^{2}$

$6 x$

$6$

$6 x^{4}$

$6 x^{3}$

$6 x^{2}$

$x^{3}$

$6 x^{2}$

$6 x$

$x^{3}$

$x^{2}$

$x$

$7 x^{2}$

$5 x$

$6$

$7 x^{2}$

$7 x$

$7$

$2 x$

$1$

Step 16

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

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