Algebra Examples

$(8 - x - 6 x^{2} + x^{3}) \div (x^{2} - 7 x + 6)$

Step 1

Expand $8 - x - 6 x^{2} + x^{3}$ .

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Step 1.1

Reorder $8$ and $- x$ .

$\frac{- x + 8 - 6 x^{2} + x^{3}}{x^{2} - 7 x + 6}$

Step 1.2

Move $8$ .

$\frac{- x - 6 x^{2} + 8 + x^{3}}{x^{2} - 7 x + 6}$

Step 1.3

Reorder $- x$ and $- 6 x^{2}$ .

$\frac{- 6 x^{2} - x + 8 + x^{3}}{x^{2} - 7 x + 6}$

Step 1.4

Move $8$ .

$\frac{- 6 x^{2} - x + x^{3} + 8}{x^{2} - 7 x + 6}$

Step 1.5

Move $- x$ .

$\frac{- 6 x^{2} + x^{3} - x + 8}{x^{2} - 7 x + 6}$

Step 1.6

Reorder $- 6 x^{2}$ and $x^{3}$ .

$\frac{x^{3} - 6 x^{2} - x + 8}{x^{2} - 7 x + 6}$

Step 2

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

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

Step 3

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

$x$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

Step 4

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

$x^{3}$

$7 x^{2}$

$6 x$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 7 x^{2} + 6 x$

$x$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

$x^{3}$

$7 x^{2}$

$6 x$

Step 6

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

$x$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

$x^{3}$

$7 x^{2}$

$6 x$

$x^{2}$

$7 x$

Step 7

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

$x$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

$x^{3}$

$7 x^{2}$

$6 x$

$x^{2}$

$7 x$

$8$

Step 8

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

$x$

$1$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

$x^{3}$

$7 x^{2}$

$6 x$

$x^{2}$

$7 x$

$8$

Step 9

Multiply the new quotient term by the divisor.

$x$

$1$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

$x^{3}$

$7 x^{2}$

$6 x$

$x^{2}$

$7 x$

$8$

$x^{2}$

$7 x$

$6$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} - 7 x + 6$

$x$

$1$

$x^{2}$

$7 x$

$6$

$x^{3}$

$6 x^{2}$

$x$

$8$

$x^{3}$

$7 x^{2}$

$6 x$

$x^{2}$

$7 x$

$8$

$x^{2}$

$7 x$

$6$

Step 11

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

						$x$	+	$1$
$x^{2}$	-	$7 x$	+	$6$		$x^{3}$	-	$6 x^{2}$	-	$x$	+	$8$
					-	$x^{3}$	+	$7 x^{2}$	-	$6 x$
							+	$x^{2}$	-	$7 x$	+	$8$
							-	$x^{2}$	+	$7 x$	-	$6$
											+	$2$

Step 12

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

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