Algebra Examples

$\frac{5 x^{4} - 2 x^{2}}{x^{3} - x^{2} + 2}$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

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

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

Step 1.2

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $x^{3}$ .

$5 x$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

Step 1.3

Multiply the new quotient term by the divisor.

$5 x$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$5 x^{4}$

$5 x^{3}$

$0$

$10 x$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} - 5 x^{3} + 0 + 10 x$

$5 x$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$5 x^{4}$

$5 x^{3}$

$0$

$10 x$

Step 1.5

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

$5 x$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$5 x^{4}$

$5 x^{3}$

$0$

$10 x$

$5 x^{3}$

$2 x^{2}$

$10 x$

Step 1.6

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

$5 x$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$5 x^{4}$

$5 x^{3}$

$0$

$10 x$

$5 x^{3}$

$2 x^{2}$

$10 x$

$0$

Step 1.7

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x^{3}$ .

$5 x$

$5$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$5 x^{4}$

$5 x^{3}$

$0$

$10 x$

$5 x^{3}$

$2 x^{2}$

$10 x$

$0$

Step 1.8

Multiply the new quotient term by the divisor.

$5 x$

$5$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$5 x^{4}$

$5 x^{3}$

$0$

$10 x$

$5 x^{3}$

$2 x^{2}$

$10 x$

$0$

$5 x^{3}$

$5 x^{2}$

$0$

$10$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 5 x^{2} + 0 + 10$

$5 x$

$5$

$x^{3}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$5 x^{4}$

$5 x^{3}$

$0$

$10 x$

$5 x^{3}$

$2 x^{2}$

$10 x$

$0$

$5 x^{3}$

$5 x^{2}$

$0$

$10$

Step 1.10

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

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

Step 1.11

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

$5 x + 5 + \frac{3 x^{2} - 10 x - 10}{x^{3} - x^{2} + 2}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$3 x^{2} - 10 x - 10$