Algebra Examples

$\frac{(36 a^{5} b^{8} c^{9} + 32 a^{8} b^{7} c^{4} - 12 a^{9} b^{3} c)}{7 a^{2} b^{2} c}$

Step 1

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

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

Step 2

Divide the highest order term in the dividend $- 12 a^{9} b^{3} c$ by the highest order term in divisor $7 a^{2} b^{2} c$ .

$\frac{12 a^{7} b}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

Step 3

Multiply the new quotient term by the divisor.

$\frac{12 a^{7} b}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 a^{9} b^{3} c + 0 + 0$

$\frac{12 a^{7} b}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

Step 5

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

$\frac{12 a^{7} b}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

Step 6

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

$\frac{12 a^{7} b}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

Step 7

Divide the highest order term in the dividend $32 a^{8} b^{7} c^{4}$ by the highest order term in divisor $7 a^{2} b^{2} c$ .

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

Step 8

Multiply the new quotient term by the divisor.

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $32 a^{8} b^{7} c^{4} + 0 + 0$

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

Step 10

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

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

Step 11

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

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

Step 12

Divide the highest order term in the dividend $36 c^{9} b^{8} a^{5}$ by the highest order term in divisor $7 a^{2} b^{2} c$ .

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$0 a^{5}$

$0 a^{4}$

$\frac{36 c^{8} b^{6} a^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

Step 13

Multiply the new quotient term by the divisor.

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$0 a^{5}$

$0 a^{4}$

$\frac{36 c^{8} b^{6} a^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$36 c^{9} b^{8} a^{5}$

$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $36 c^{9} b^{8} a^{5} + 0 + 0$

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$0 a^{5}$

$0 a^{4}$

$\frac{36 c^{8} b^{6} a^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$36 c^{9} b^{8} a^{5}$

$0$

Step 15

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

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$0 a^{5}$

$0 a^{4}$

$\frac{36 c^{8} b^{6} a^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$36 c^{9} b^{8} a^{5}$

$0$

Step 16

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

$\frac{12 a^{7} b}{7}$

$\frac{32 a^{6} b^{5} c^{3}}{7}$

$0 a^{5}$

$0 a^{4}$

$\frac{36 c^{8} b^{6} a^{3}}{7}$

$7 a^{2} b^{2} c$

$0 a$

$0$

$12 a^{9} b^{3} c$

$32 a^{8} b^{7} c^{4}$

$0 a^{7}$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$0$

$12 a^{9} b^{3} c$

$0$

$32 a^{8} b^{7} c^{4}$

$0$

$0 a^{6}$

$36 c^{9} b^{8} a^{5}$

$32 a^{8} b^{7} c^{4}$

$0$

$36 c^{9} b^{8} a^{5}$

$0 a^{4}$

$0 a^{3}$

$36 c^{9} b^{8} a^{5}$

$0$

$0 a^{2}$

$0 a$

$0$

Step 17

Since the remander is $0$ , the final answer is the quotient.

$- \frac{12 a^{7} b}{7} + \frac{32 a^{6} b^{5} c^{3}}{7} + 0 a^{5} + 0 a^{4} + \frac{36 c^{8} b^{6} a^{3}}{7}$