Algebra Examples

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

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 $0$ .

$x^{7}$

$0 x^{6}$

$x^{5}$

$0 x^{4}$

$x^{3}$

$x^{2}$

$0 x$

$0$

$2 x^{7}$

$0 x^{6}$

$3 x^{5}$

$0 x^{4}$

$5 x^{3}$

$7 x^{2}$

$0 x$

$0$

Step 1.2

Divide the highest order term in the dividend $2 x^{7}$ by the highest order term in divisor $x^{7}$ .

$2$

$x^{7}$

$0 x^{6}$

$x^{5}$

$0 x^{4}$

$x^{3}$

$x^{2}$

$0 x$

$0$

$2 x^{7}$

$0 x^{6}$

$3 x^{5}$

$0 x^{4}$

$5 x^{3}$

$7 x^{2}$

$0 x$

$0$

Step 1.3

Multiply the new quotient term by the divisor.

$2$

$x^{7}$

$0 x^{6}$

$x^{5}$

$0 x^{4}$

$x^{3}$

$x^{2}$

$0 x$

$0$

$2 x^{7}$

$0 x^{6}$

$3 x^{5}$

$0 x^{4}$

$5 x^{3}$

$7 x^{2}$

$0 x$

$0$

$2 x^{7}$

$0$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$0$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{7} + 0 + 2 x^{5} + 0 + 2 x^{3} + 2 x^{2} + 0 + 0$

$2$

$x^{7}$

$0 x^{6}$

$x^{5}$

$0 x^{4}$

$x^{3}$

$x^{2}$

$0 x$

$0$

$2 x^{7}$

$0 x^{6}$

$3 x^{5}$

$0 x^{4}$

$5 x^{3}$

$7 x^{2}$

$0 x$

$0$

$2 x^{7}$

$0$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$0$

Step 1.5

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

$2$

$x^{7}$

$0 x^{6}$

$x^{5}$

$0 x^{4}$

$x^{3}$

$x^{2}$

$0 x$

$0$

$2 x^{7}$

$0 x^{6}$

$3 x^{5}$

$0 x^{4}$

$5 x^{3}$

$7 x^{2}$

$0 x$

$0$

$2 x^{7}$

$0$

$2 x^{5}$

$0$

$2 x^{3}$

$2 x^{2}$

$0$

$5 x^{5}$

$0$

$3 x^{3}$

$9 x^{2}$

$0$

Step 1.6

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

$2 + \frac{- 5 x^{5} + 3 x^{3} - 9 x^{2}}{x^{7} + x^{5} + x^{3} + x^{2}}$

Step 2

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

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