Algebra Examples

$(2 x^{4} + 2 x^{3} - 10 x - 9) \div (x^{3} + x^{2} - 5)$

Step 1

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

$x^{3}$

$x^{2}$

$0 x$

$5$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$10 x$

$9$

Step 2

Divide the highest order term in the dividend $2 x^{4}$ by the highest order term in divisor $x^{3}$ .

$2 x$

$x^{3}$

$x^{2}$

$0 x$

$5$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$10 x$

$9$

Step 3

Multiply the new quotient term by the divisor.

$2 x$

$x^{3}$

$x^{2}$

$0 x$

$5$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$10 x$

$9$

$2 x^{4}$

$2 x^{3}$

$0$

$10 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 2 x^{3} + 0 - 10 x$

$2 x$

$x^{3}$

$x^{2}$

$0 x$

$5$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$10 x$

$9$

$2 x^{4}$

$2 x^{3}$

$0$

$10 x$

Step 5

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

$2 x$

$x^{3}$

$x^{2}$

$0 x$

$5$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$10 x$

$9$

$2 x^{4}$

$2 x^{3}$

$0$

$10 x$

$0$

Step 6

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

								$2 x$
$x^{3}$	+	$x^{2}$	+	$0 x$	-	$5$		$2 x^{4}$	+	$2 x^{3}$	+	$0 x^{2}$	-	$10 x$	-	$9$
							-	$2 x^{4}$	-	$2 x^{3}$	-	$0$	+	$10 x$
														$0$	-	$9$

Step 7

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

$2 x - \frac{9}{x^{3} + x^{2} - 5}$