Algebra Examples

$\frac{(x^{4} + 36)}{(x^{2} - 8)}$

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 $0$ .

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

Step 1.2

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

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 - 8 x^{2}$

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

Step 1.5

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

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

Step 1.6

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

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

$0 x$

$36$

Step 1.7

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$0 x$

$8$

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

$0 x$

$36$

Step 1.8

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$8$

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

$0 x$

$36$

$8 x^{2}$

$0$

$64$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 0 - 64$

$x^{2}$

$0 x$

$8$

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

$0 x$

$36$

$8 x^{2}$

$0$

$64$

Step 1.10

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

$x^{2}$

$0 x$

$8$

$x^{2}$

$0 x$

$8$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$36$

$x^{4}$

$0$

$8 x^{2}$

$0 x$

$36$

$8 x^{2}$

$0$

$64$

$100$

Step 1.11

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

$x^{2} + 8 + \frac{100}{x^{2} - 8}$

Step 2

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

$100$