Algebra Examples

$\frac{- 3 d^{8} f^{6} + 27 d^{5} f^{8} - 15 d^{5} f^{6}}{- 3 d^{5} f^{6}}$

Step 1

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

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

Step 2

Divide the highest order term in the dividend $- 3 d^{8} f^{6}$ by the highest order term in divisor $- 3 f^{6} d^{5}$ .

$d^{3}$

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

Step 3

Multiply the new quotient term by the divisor.

$d^{3}$

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 d^{8} f^{6} + 0 + 0 + 0 + 0 + 0$

$d^{3}$

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0$

Step 5

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

$d^{3}$

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0$

Step 6

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

$d^{3}$

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0$

$0 d^{2}$

$0 d$

$0$

Step 7

Divide the highest order term in the dividend $27 f^{8} d^{5}$ by the highest order term in divisor $- 3 f^{6} d^{5}$ .

$d^{3}$

$0 d^{2}$

$0 d$

$9 f^{2}$

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0$

$0 d^{2}$

$0 d$

$0$

Step 8

Multiply the new quotient term by the divisor.

$d^{3}$

$0 d^{2}$

$0 d$

$9 f^{2}$

$3 f^{6} d^{5}$

$0 d^{4}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0 d^{7}$

$0 d^{6}$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0 d^{3}$

$0 d^{2}$

$0 d$

$0$

$3 d^{8} f^{6}$

$0$

$27 f^{8} d^{5}$

$15 f^{6} d^{5}$

$0$

$0 d^{2}$

$0 d$

$0$

$27 f^{8} d^{5}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $27 f^{8} d^{5} + 0 + 0 + 0 + 0 + 0$

													$d^{3}$	+	$0 d^{2}$	+	$0 d$	-	$9 f^{2}$
-	$3 f^{6} d^{5}$	+	$0 d^{4}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$	-	$3 d^{8} f^{6}$	+	$0 d^{7}$	+	$0 d^{6}$	+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$
												+	$3 d^{8} f^{6}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																		+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0$	+	$0 d^{2}$	+	$0 d$	+	$0$
																		-	$27 f^{8} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$

Step 10

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

													$d^{3}$	+	$0 d^{2}$	+	$0 d$	-	$9 f^{2}$
-	$3 f^{6} d^{5}$	+	$0 d^{4}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$	-	$3 d^{8} f^{6}$	+	$0 d^{7}$	+	$0 d^{6}$	+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$
												+	$3 d^{8} f^{6}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																		+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0$	+	$0 d^{2}$	+	$0 d$	+	$0$
																		-	$27 f^{8} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																				-	$15 f^{6} d^{5}$	+	$0$	+	$0$	+	$0$	+	$0$

Step 11

Divide the highest order term in the dividend $- 15 f^{6} d^{5}$ by the highest order term in divisor $- 3 f^{6} d^{5}$ .

													$d^{3}$	+	$0 d^{2}$	+	$0 d$	-	$9 f^{2}$	+	$5$
-	$3 f^{6} d^{5}$	+	$0 d^{4}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$	-	$3 d^{8} f^{6}$	+	$0 d^{7}$	+	$0 d^{6}$	+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$
												+	$3 d^{8} f^{6}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																		+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0$	+	$0 d^{2}$	+	$0 d$	+	$0$
																		-	$27 f^{8} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																				-	$15 f^{6} d^{5}$	+	$0$	+	$0$	+	$0$	+	$0$

Step 12

Multiply the new quotient term by the divisor.

													$d^{3}$	+	$0 d^{2}$	+	$0 d$	-	$9 f^{2}$	+	$5$
-	$3 f^{6} d^{5}$	+	$0 d^{4}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$	-	$3 d^{8} f^{6}$	+	$0 d^{7}$	+	$0 d^{6}$	+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$
												+	$3 d^{8} f^{6}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																		+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0$	+	$0 d^{2}$	+	$0 d$	+	$0$
																		-	$27 f^{8} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																				-	$15 f^{6} d^{5}$	+	$0$	+	$0$	+	$0$	+	$0$
																				-	$15 f^{6} d^{5}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0$

Step 13

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 f^{6} d^{5} + 0 + 0 + 0 + 0 + 0$

													$d^{3}$	+	$0 d^{2}$	+	$0 d$	-	$9 f^{2}$	+	$5$
-	$3 f^{6} d^{5}$	+	$0 d^{4}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$	-	$3 d^{8} f^{6}$	+	$0 d^{7}$	+	$0 d^{6}$	+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$
												+	$3 d^{8} f^{6}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																		+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0$	+	$0 d^{2}$	+	$0 d$	+	$0$
																		-	$27 f^{8} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																				-	$15 f^{6} d^{5}$	+	$0$	+	$0$	+	$0$	+	$0$
																				+	$15 f^{6} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$

Step 14

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

													$d^{3}$	+	$0 d^{2}$	+	$0 d$	-	$9 f^{2}$	+	$5$
-	$3 f^{6} d^{5}$	+	$0 d^{4}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$	-	$3 d^{8} f^{6}$	+	$0 d^{7}$	+	$0 d^{6}$	+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0 d^{3}$	+	$0 d^{2}$	+	$0 d$	+	$0$
												+	$3 d^{8} f^{6}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																		+	$27 f^{8} d^{5}$	-	$15 f^{6} d^{5}$	+	$0$	+	$0 d^{2}$	+	$0 d$	+	$0$
																		-	$27 f^{8} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																				-	$15 f^{6} d^{5}$	+	$0$	+	$0$	+	$0$	+	$0$
																				+	$15 f^{6} d^{5}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																															$0$

Step 15

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

$d^{3} + 0 d^{2} + 0 d - 9 f^{2} + 5$