Algebra Examples

$\frac{4 x^{5} - x^{3} + 7}{2 x^{5} + x^{2}}$

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 $0$ .

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$7$

Step 1.2

Divide the highest order term in the dividend $4 x^{5}$ by the highest order term in divisor $2 x^{5}$ .

$2$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$7$

Step 1.3

Multiply the new quotient term by the divisor.

$2$

$2 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$4 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$7$

$4 x^{5}$

$0$

$2 x^{2}$

$0$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{5} + 0 + 0 + 2 x^{2} + 0 + 0$

												$2$
$2 x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	+	$x^{2}$	+	$0 x$	+	$0$		$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$7$
											-	$4 x^{5}$	-	$0$	-	$0$	-	$2 x^{2}$	-	$0$	-	$0$

Step 1.5

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

												$2$
$2 x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	+	$x^{2}$	+	$0 x$	+	$0$		$4 x^{5}$	+	$0 x^{4}$	-	$x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$7$
											-	$4 x^{5}$	-	$0$	-	$0$	-	$2 x^{2}$	-	$0$	-	$0$
															-	$x^{3}$	-	$2 x^{2}$	+	$0$	+	$7$

Step 1.6

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

$2 + \frac{- x^{3} - 2 x^{2} + 7}{2 x^{5} + x^{2}}$

Step 2

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

$- x^{3} - 2 x^{2} + 7$