Simplify 1/(a-b)+4/(b-a)-8/(a+b)-(11a-5b)/(b^2-a^2)

$\frac{1}{a - b} + \frac{4}{b - a} - \frac{8}{a + b} - \frac{11 a - 5 b}{b^{2} - a^{2}}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = b$ and $b = a$ .

$\frac{1}{a - b} + \frac{4}{b - a} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 2

Simplify terms.

Factor $- 1$ out of $b$ .

$\frac{1}{a - b} + \frac{4}{- 1 (- b) - a} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Factor $- 1$ out of $- a$ .

$\frac{1}{a - b} + \frac{4}{- 1 (- b) - (a)} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Factor $- 1$ out of $- 1 (- b) - (a)$ .

$\frac{1}{a - b} + \frac{4}{- 1 (- b + a)} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Simplify the expression.

Move a negative from the denominator of $\frac{4}{- 1 (- b + a)}$ to the numerator.

$\frac{1}{a - b} + \frac{- 1 \cdot 4}{- b + a} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Reorder terms.

$\frac{1}{a - b} + \frac{- 1 \cdot 4}{a - b} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Combine the numerators over the common denominator.

$\frac{1 - 1 \cdot 4}{a - b} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 3

Simplify the numerator.

Multiply $- 1$ by $4$ .

$\frac{1 - 4}{a - b} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Subtract $4$ from $1$ .

$\frac{- 3}{a - b} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 4

Move the negative in front of the fraction.

$- \frac{3}{a - b} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 5

To write $- \frac{3}{a - b}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$- \frac{3}{a - b} \cdot \frac{a + b}{a + b} - \frac{8}{a + b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 6

To write $- \frac{8}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a - b}{a - b}$ .

$- \frac{3}{a - b} \cdot \frac{a + b}{a + b} - \frac{8}{a + b} \cdot \frac{a - b}{a - b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 7

Write each expression with a common denominator of $(a - b) (a + b)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{a - b}$ by $\frac{a + b}{a + b}$ .

$- \frac{3 (a + b)}{(a - b) (a + b)} - \frac{8}{a + b} \cdot \frac{a - b}{a - b} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Multiply $\frac{8}{a + b}$ by $\frac{a - b}{a - b}$ .

$- \frac{3 (a + b)}{(a - b) (a + b)} - \frac{8 (a - b)}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Reorder the factors of $(a - b) (a + b)$ .

$- \frac{3 (a + b)}{(a + b) (a - b)} - \frac{8 (a - b)}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 8

Combine the numerators over the common denominator.

$\frac{- 3 (a + b) - 8 (a - b)}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 9

Simplify each term.

Simplify the numerator.

Factor $- 1$ out of $- 3 (a + b) - 8 (a - b)$ .

Factor $- 1$ out of $- 3 (a + b)$ .

$\frac{- (3 (a + b)) - 8 (a - b)}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Factor $- 1$ out of $- 8 (a - b)$ .

$\frac{- (3 (a + b)) - (8 (a - b))}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Factor $- 1$ out of $- (3 (a + b)) - (8 (a - b))$ .

$\frac{- (3 (a + b) + 8 (a - b))}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Apply the distributive property.

$\frac{- (3 a + 3 b + 8 (a - b))}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Apply the distributive property.

$\frac{- (3 a + 3 b + 8 a + 8 (- b))}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Multiply $- 1$ by $8$ .

$\frac{- (3 a + 3 b + 8 a - 8 b)}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Add $3 a$ and $8 a$ .

$\frac{- (11 a + 3 b - 8 b)}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Subtract $8 b$ from $3 b$ .

$\frac{- (11 a - 5 b)}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Move the negative in front of the fraction.

$- \frac{11 a - 5 b}{(a + b) (a - b)} - \frac{11 a - 5 b}{(b + a) (b - a)}$

Step 10

Simplify with factoring out.

Reorder terms.

$- \frac{11 a - 5 b}{(a + b) (a - b)} - \frac{11 a - 5 b}{(a + b) (b - a)}$

Factor $- 1$ out of $b$ .

$- \frac{11 a - 5 b}{(a + b) (a - b)} - \frac{11 a - 5 b}{(a + b) (- 1 (- b) - a)}$

Factor $- 1$ out of $- a$ .

$- \frac{11 a - 5 b}{(a + b) (a - b)} - \frac{11 a - 5 b}{(a + b) (- 1 (- b) - (a))}$

Factor $- 1$ out of $- 1 (- b) - (a)$ .

$- \frac{11 a - 5 b}{(a + b) (a - b)} - \frac{11 a - 5 b}{(a + b) (- 1 (- b + a))}$

Reorder terms.

$- \frac{11 a - 5 b}{(a + b) (a - b)} - \frac{11 a - 5 b}{(a + b) (- 1 (a - b))}$

Step 11

To write $- \frac{11 a - 5 b}{(a + b) (a - b)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$- \frac{11 a - 5 b}{(a + b) (a - b)} \cdot \frac{- 1}{- 1} - \frac{11 a - 5 b}{(a + b) (- 1 (a - b))}$

Step 12

Write each expression with a common denominator of $(a + b) (a - b) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{11 a - 5 b}{(a + b) (a - b)}$ by $\frac{- 1}{- 1}$ .

$- \frac{(11 a - 5 b) \cdot - 1}{(a + b) (a - b) \cdot - 1} - \frac{11 a - 5 b}{(a + b) (- 1 (a - b))}$

Reorder the factors of $(a + b) (a - b) \cdot - 1$ .

$- \frac{(11 a - 5 b) \cdot - 1}{- (a + b) (a - b)} - \frac{11 a - 5 b}{(a + b) (- 1 (a - b))}$

Reorder the factors of $(a + b) (- 1 (a - b))$ .

$- \frac{(11 a - 5 b) \cdot - 1}{- (a + b) (a - b)} - \frac{11 a - 5 b}{- (a + b) (a - b)}$

Step 13

Combine the numerators over the common denominator.

$\frac{- (11 a - 5 b) \cdot - 1 - (11 a - 5 b)}{- (a + b) (a - b)}$

Step 14

Simplify the numerator.

Factor $- (11 a - 5 b)$ out of $- (11 a - 5 b) \cdot - 1 - (11 a - 5 b)$ .

Move $11 a - 5 b$ .

$\frac{- 1 \cdot - 1 (11 a - 5 b) - (11 a - 5 b)}{- (a + b) (a - b)}$

Factor $- (11 a - 5 b)$ out of $- 1 \cdot - 1 (11 a - 5 b)$ .

$\frac{- (11 a - 5 b) (- 1) - (11 a - 5 b)}{- (a + b) (a - b)}$

Factor $- (11 a - 5 b)$ out of $- (11 a - 5 b)$ .

$\frac{- (11 a - 5 b) (- 1) - (11 a - 5 b) (1)}{- (a + b) (a - b)}$

Factor $- (11 a - 5 b)$ out of $- (11 a - 5 b) (- 1) - (11 a - 5 b) (1)$ .

$\frac{- (11 a - 5 b) (- 1 + 1)}{- (a + b) (a - b)}$

Add $- 1$ and $1$ .

$\frac{- (11 a - 5 b) \cdot 0}{- (a + b) (a - b)}$

Combine exponents.

Multiply $0$ by $- 1$ .

$\frac{0 (11 a - 5 b)}{- (a + b) (a - b)}$

Multiply $0$ by $11 a - 5 b$ .

$\frac{0}{- (a + b) (a - b)}$

Step 15

Divide $0$ by $- (a + b) (a - b)$ .

$0$

Simplify 1/(a-b)+4/(b-a)-8/(a+b)-(11a-5b)/(b^2-a^2)

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