Pre-Algebra Examples

$\frac{{(x - 5)}^{6} (x + 7)}{x^{2} + 8 x + 7} \div \frac{x^{21} - 10 x + 25}{x^{2} + 14 x + 49}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{{(x - 5)}^{6} (x + 7)}{x^{2} + 8 x + 7} \cdot \frac{x^{2} + 14 x + 49}{x^{21} - 10 x + 25}$

Step 2

Factor $x^{2} + 8 x + 7$ using the AC method.

Tap for more steps...

Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $7$ and whose sum is $8$ .

$1, 7$

Step 2.2

Write the factored form using these integers.

$\frac{{(x - 5)}^{6} (x + 7)}{(x + 1) (x + 7)} \cdot \frac{x^{2} + 14 x + 49}{x^{21} - 10 x + 25}$

Step 3

Cancel the common factor of $x + 7$ .

Tap for more steps...

Step 3.1

Cancel the common factor.

$\frac{{(x - 5)}^{6} (x + 7)}{(x + 1) (x + 7)} \cdot \frac{x^{2} + 14 x + 49}{x^{21} - 10 x + 25}$

Step 3.2

Rewrite the expression.

$\frac{{(x - 5)}^{6}}{x + 1} \cdot \frac{x^{2} + 14 x + 49}{x^{21} - 10 x + 25}$

Step 4

Factor using the perfect square rule.

Tap for more steps...

Step 4.1

Rewrite $49$ as $7^{2}$ .

$\frac{{(x - 5)}^{6}}{x + 1} \cdot \frac{x^{2} + 14 x + 7^{2}}{x^{21} - 10 x + 25}$

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 4.3

Rewrite the polynomial.

$\frac{{(x - 5)}^{6}}{x + 1} \cdot \frac{x^{2} + 2 \cdot x \cdot 7 + 7^{2}}{x^{21} - 10 x + 25}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 7$ .

$\frac{{(x - 5)}^{6}}{x + 1} \cdot \frac{{(x + 7)}^{2}}{x^{21} - 10 x + 25}$

Step 5

Multiply $\frac{{(x - 5)}^{6}}{x + 1}$ by $\frac{{(x + 7)}^{2}}{x^{21} - 10 x + 25}$ .

$\frac{{(x - 5)}^{6} {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 6

Use the Binomial Theorem.

$\frac{(x^{6} + 6 x^{5} \cdot - 5 + 15 x^{4} {(- 5)}^{2} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7

Simplify each term.

Tap for more steps...

Step 7.1

Multiply $- 5$ by $6$ .

$\frac{(x^{6} - 30 x^{5} + 15 x^{4} {(- 5)}^{2} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.2

Raise $- 5$ to the power of $2$ .

$\frac{(x^{6} - 30 x^{5} + 15 x^{4} \cdot 25 + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.3

Multiply $25$ by $15$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} + 20 x^{3} {(- 5)}^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.4

Raise $- 5$ to the power of $3$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} + 20 x^{3} \cdot - 125 + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.5

Multiply $- 125$ by $20$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 15 x^{2} {(- 5)}^{4} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.6

Raise $- 5$ to the power of $4$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 15 x^{2} \cdot 625 + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.7

Multiply $625$ by $15$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 6 x {(- 5)}^{5} + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.8

Raise $- 5$ to the power of $5$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} + 6 x \cdot - 3125 + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.9

Multiply $- 3125$ by $6$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + {(- 5)}^{6}) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 7.10

Raise $- 5$ to the power of $6$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) {(x + 7)}^{2}}{(x + 1) (x^{21} - 10 x + 25)}$

Step 8

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) ((x + 7) (x + 7))}{(x + 1) (x^{21} - 10 x + 25)}$

Step 9

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Tap for more steps...

Step 9.1

Apply the distributive property.

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x (x + 7) + 7 (x + 7))}{(x + 1) (x^{21} - 10 x + 25)}$

Step 9.2

Apply the distributive property.

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x \cdot x + x \cdot 7 + 7 (x + 7))}{(x + 1) (x^{21} - 10 x + 25)}$

Step 9.3

Apply the distributive property.

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7)}{(x + 1) (x^{21} - 10 x + 25)}$

Step 10

Simplify and combine like terms.

Tap for more steps...

Step 10.1

Simplify each term.

Tap for more steps...

Step 10.1.1

Multiply $x$ by $x$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x^{2} + x \cdot 7 + 7 x + 7 \cdot 7)}{(x + 1) (x^{21} - 10 x + 25)}$

Step 10.1.2

Move $7$ to the left of $x$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x^{2} + 7 \cdot x + 7 x + 7 \cdot 7)}{(x + 1) (x^{21} - 10 x + 25)}$

Step 10.1.3

Multiply $7$ by $7$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x^{2} + 7 x + 7 x + 49)}{(x + 1) (x^{21} - 10 x + 25)}$

Step 10.2

Add $7 x$ and $7 x$ .

$\frac{(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x^{2} + 14 x + 49)}{(x + 1) (x^{21} - 10 x + 25)}$

Step 11

Expand $(x^{6} - 30 x^{5} + 375 x^{4} - 2500 x^{3} + 9375 x^{2} - 18750 x + 15625) (x^{2} + 14 x + 49)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{6} x^{2} + x^{6} (14 x) + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12

Simplify each term.

Tap for more steps...

Step 12.1

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 12.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{6 + 2} + x^{6} (14 x) + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.1.2

Add $6$ and $2$ .

$\frac{x^{8} + x^{6} (14 x) + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.2

Rewrite using the commutative property of multiplication.

$\frac{x^{8} + 14 x^{6} x + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.3

Multiply $x^{6}$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.3.1

Move $x$ .

$\frac{x^{8} + 14 (x \cdot x^{6}) + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.3.2

Multiply $x$ by $x^{6}$ .

Tap for more steps...

Step 12.3.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{8} + 14 (x^{1} x^{6}) + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{1 + 6} + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.3.3

Add $1$ and $6$ .

$\frac{x^{8} + 14 x^{7} + x^{6} \cdot 49 - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.4

Move $49$ to the left of $x^{6}$ .

$\frac{x^{8} + 14 x^{7} + 49 \cdot x^{6} - 30 x^{5} x^{2} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.5

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 12.5.1

Move $x^{2}$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 (x^{2} x^{5}) - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{2 + 5} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.5.3

Add $2$ and $5$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 30 x^{5} (14 x) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.6

Rewrite using the commutative property of multiplication.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 30 \cdot 14 x^{5} x - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.7

Multiply $x^{5}$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.7.1

Move $x$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 30 \cdot 14 (x \cdot x^{5}) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.7.2

Multiply $x$ by $x^{5}$ .

Tap for more steps...

Step 12.7.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 30 \cdot 14 (x^{1} x^{5}) - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 30 \cdot 14 x^{1 + 5} - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.7.3

Add $1$ and $5$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 30 \cdot 14 x^{6} - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.8

Multiply $- 30$ by $14$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 30 x^{5} \cdot 49 + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.9

Multiply $49$ by $- 30$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{4} x^{2} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.10

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 12.10.1

Move $x^{2}$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 (x^{2} x^{4}) + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.10.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{2 + 4} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.10.3

Add $2$ and $4$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 375 x^{4} (14 x) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.11

Rewrite using the commutative property of multiplication.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 375 \cdot 14 x^{4} x + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.12

Multiply $x^{4}$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.12.1

Move $x$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 375 \cdot 14 (x \cdot x^{4}) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.12.2

Multiply $x$ by $x^{4}$ .

Tap for more steps...

Step 12.12.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 375 \cdot 14 (x^{1} x^{4}) + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.12.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 375 \cdot 14 x^{1 + 4} + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.12.3

Add $1$ and $4$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 375 \cdot 14 x^{5} + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.13

Multiply $375$ by $14$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 375 x^{4} \cdot 49 - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.14

Multiply $49$ by $375$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{3} x^{2} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.15

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 12.15.1

Move $x^{2}$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 (x^{2} x^{3}) - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.15.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{2 + 3} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.15.3

Add $2$ and $3$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 2500 x^{3} (14 x) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.16

Rewrite using the commutative property of multiplication.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 2500 \cdot 14 x^{3} x - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.17

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.17.1

Move $x$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 2500 \cdot 14 (x \cdot x^{3}) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.17.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 12.17.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 2500 \cdot 14 (x^{1} x^{3}) - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.17.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 2500 \cdot 14 x^{1 + 3} - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.17.3

Add $1$ and $3$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 2500 \cdot 14 x^{4} - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.18

Multiply $- 2500$ by $14$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 2500 x^{3} \cdot 49 + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.19

Multiply $49$ by $- 2500$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{2} x^{2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.20

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 12.20.1

Move $x^{2}$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 (x^{2} x^{2}) + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.20.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{2 + 2} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.20.3

Add $2$ and $2$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 9375 x^{2} (14 x) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.21

Rewrite using the commutative property of multiplication.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 9375 \cdot 14 x^{2} x + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.22

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.22.1

Move $x$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 9375 \cdot 14 (x \cdot x^{2}) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.22.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 12.22.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 9375 \cdot 14 (x^{1} x^{2}) + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.22.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 9375 \cdot 14 x^{1 + 2} + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.22.3

Add $1$ and $2$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 9375 \cdot 14 x^{3} + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.23

Multiply $9375$ by $14$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 9375 x^{2} \cdot 49 - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.24

Multiply $49$ by $9375$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x \cdot x^{2} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.25

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 12.25.1

Move $x^{2}$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 (x^{2} x) - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.25.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 12.25.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 (x^{2} x^{1}) - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.25.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{2 + 1} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.25.3

Add $2$ and $1$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 18750 x (14 x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.26

Rewrite using the commutative property of multiplication.

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 18750 \cdot 14 x \cdot x - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.27

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.27.1

Move $x$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 18750 \cdot 14 (x \cdot x) - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.27.2

Multiply $x$ by $x$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 18750 \cdot 14 x^{2} - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.28

Multiply $- 18750$ by $14$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 18750 x \cdot 49 + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.29

Multiply $49$ by $- 18750$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 15625 (14 x) + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.30

Multiply $14$ by $15625$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 15625 \cdot 49}{(x + 1) (x^{21} - 10 x + 25)}$

Step 12.31

Multiply $15625$ by $49$ .

$\frac{x^{8} + 14 x^{7} + 49 x^{6} - 30 x^{7} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 13

Subtract $30 x^{7}$ from $14 x^{7}$ .

$\frac{x^{8} - 16 x^{7} + 49 x^{6} - 420 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 14

Subtract $420 x^{6}$ from $49 x^{6}$ .

$\frac{x^{8} - 16 x^{7} - 371 x^{6} - 1470 x^{5} + 375 x^{6} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 15

Add $- 371 x^{6}$ and $375 x^{6}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} - 1470 x^{5} + 5250 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 16

Add $- 1470 x^{5}$ and $5250 x^{5}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 3780 x^{5} + 18375 x^{4} - 2500 x^{5} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 17

Subtract $2500 x^{5}$ from $3780 x^{5}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} + 18375 x^{4} - 35000 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 18

Subtract $35000 x^{4}$ from $18375 x^{4}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 16625 x^{4} - 122500 x^{3} + 9375 x^{4} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 19

Add $- 16625 x^{4}$ and $9375 x^{4}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 122500 x^{3} + 131250 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 20

Add $- 122500 x^{3}$ and $131250 x^{3}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} + 8750 x^{3} + 459375 x^{2} - 18750 x^{3} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 21

Subtract $18750 x^{3}$ from $8750 x^{3}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 459375 x^{2} - 262500 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 22

Subtract $262500 x^{2}$ from $459375 x^{2}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 196875 x^{2} - 918750 x + 15625 x^{2} + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 23

Add $196875 x^{2}$ and $15625 x^{2}$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 212500 x^{2} - 918750 x + 218750 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 24

Add $- 918750 x$ and $218750 x$ .

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 212500 x^{2} - 700000 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 25

Split the fraction $\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 212500 x^{2} - 700000 x + 765625}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 212500 x^{2} - 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 26

Split the fraction $\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 212500 x^{2} - 700000 x}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 27

Split the fraction $\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3} + 212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 28

Split the fraction $\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4} - 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 29

Split the fraction $\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5} - 7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 30

Split the fraction $\frac{x^{8} - 16 x^{7} + 4 x^{6} + 1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8} - 16 x^{7} + 4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 31

Split the fraction $\frac{x^{8} - 16 x^{7} + 4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8} - 16 x^{7}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 32

Split the fraction $\frac{x^{8} - 16 x^{7}}{(x + 1) (x^{21} - 10 x + 25)}$ into two fractions.

$\frac{x^{8}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 16 x^{7}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 33

Move the negative in front of the fraction.

$\frac{x^{8}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{16 x^{7}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 34

Move the negative in front of the fraction.

$\frac{x^{8}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{16 x^{7}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 35

Move the negative in front of the fraction.

$\frac{x^{8}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{16 x^{7}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{- 700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$

Step 36

Move the negative in front of the fraction.

$\frac{x^{8}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{16 x^{7}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{4 x^{6}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{1280 x^{5}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{7250 x^{4}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{10000 x^{3}}{(x + 1) (x^{21} - 10 x + 25)} + \frac{212500 x^{2}}{(x + 1) (x^{21} - 10 x + 25)} - \frac{700000 x}{(x + 1) (x^{21} - 10 x + 25)} + \frac{765625}{(x + 1) (x^{21} - 10 x + 25)}$