Algebra Examples

$\frac{3 v^{3} + v^{2} + 2 v}{9 v^{3}}$

Step 1

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

$9 v^{3}$

$0 v^{2}$

$0 v$

$0$

$3 v^{3}$

$v^{2}$

$2 v$

$0$

Step 2

Divide the highest order term in the dividend $3 v^{3}$ by the highest order term in divisor $9 v^{3}$ .

$\frac{1}{3}$

$9 v^{3}$

$0 v^{2}$

$0 v$

$0$

$3 v^{3}$

$v^{2}$

$2 v$

$0$

Step 3

Multiply the new quotient term by the divisor.

$\frac{1}{3}$

$9 v^{3}$

$0 v^{2}$

$0 v$

$0$

$3 v^{3}$

$v^{2}$

$2 v$

$0$

$3 v^{3}$

$0$

Step 4

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

$\frac{1}{3}$

$9 v^{3}$

$0 v^{2}$

$0 v$

$0$

$3 v^{3}$

$v^{2}$

$2 v$

$0$

$3 v^{3}$

$0$

Step 5

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

$\frac{1}{3}$

$9 v^{3}$

$0 v^{2}$

$0 v$

$0$

$3 v^{3}$

$v^{2}$

$2 v$

$0$

$3 v^{3}$

$0$

$v^{2}$

$2 v$

$0$

Step 6

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

$\frac{1}{3} + \frac{v^{2} + 2 v}{9 v^{3}}$