Algebra Examples

$\frac{14 x^{6} + 6 x^{4} - 5 x^{2}}{7 x^{5} + 2 x^{3} - 3 x}$

Step 1

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

$7 x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$3 x$

$0$

$14 x^{6}$

$0 x^{5}$

$6 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $14 x^{6}$ by the highest order term in divisor $7 x^{5}$ .

$2 x$

$7 x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$3 x$

$0$

$14 x^{6}$

$0 x^{5}$

$6 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$2 x$

$7 x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$3 x$

$0$

$14 x^{6}$

$0 x^{5}$

$6 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$0$

$14 x^{6}$

$0$

$4 x^{4}$

$0$

$6 x^{2}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $14 x^{6} + 0 + 4 x^{4} + 0 - 6 x^{2} + 0$

$2 x$

$7 x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$3 x$

$0$

$14 x^{6}$

$0 x^{5}$

$6 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$0$

$14 x^{6}$

$0$

$4 x^{4}$

$0$

$6 x^{2}$

$0$

Step 5

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

$2 x$

$7 x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$3 x$

$0$

$14 x^{6}$

$0 x^{5}$

$6 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$0$

$14 x^{6}$

$0$

$4 x^{4}$

$0$

$6 x^{2}$

$0$

$2 x^{4}$

$0$

$x^{2}$

$0$

Step 6

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

$2 x$

$7 x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$3 x$

$0$

$14 x^{6}$

$0 x^{5}$

$6 x^{4}$

$0 x^{3}$

$5 x^{2}$

$0 x$

$0$

$14 x^{6}$

$0$

$4 x^{4}$

$0$

$6 x^{2}$

$0$

$2 x^{4}$

$0$

$x^{2}$

$0$

Step 7

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

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