Algebra Examples

$\frac{5 x^{9} - 8 x^{5} + 4 x^{3} + 8 x}{x^{4}}$

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 $0$ .

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

Step 1.2

Divide the highest order term in the dividend $5 x^{9}$ by the highest order term in divisor $x^{4}$ .

$5 x^{5}$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

Step 1.3

Multiply the new quotient term by the divisor.

$5 x^{5}$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{9} + 0 + 0 + 0 + 0$

$5 x^{5}$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

Step 1.5

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

$5 x^{5}$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

$8 x^{5}$

Step 1.6

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

$5 x^{5}$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

Step 1.7

Divide the highest order term in the dividend $- 8 x^{5}$ by the highest order term in divisor $x^{4}$ .

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

Step 1.8

Multiply the new quotient term by the divisor.

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$8 x^{5}$

$0$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{5} + 0 + 0 + 0 + 0$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$8 x^{5}$

$0$

Step 1.10

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

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$8 x^{5}$

$0$

$4 x^{3}$

$0$

$8 x$

Step 1.11

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

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$5 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$0$

$5 x^{9}$

$0$

$8 x^{5}$

$0 x^{4}$

$4 x^{3}$

$0 x^{2}$

$8 x$

$8 x^{5}$

$0$

$4 x^{3}$

$0$

$8 x$

$0$

Step 1.12

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

$5 x^{5} - 8 x + \frac{4 x^{3} + 8 x}{x^{4}}$

Step 2

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

$4 x^{3} + 8 x$