express-in-partial-fraction-3-x-1-x-2-4-

Question Number 60735 by readone97 last updated on 25/May/19

e x p r e s s i n p a r t i a l f r a c t i o n 3 / (x + 1) (x^{2} - 4)

Answered by Forkum Michael Choungong last updated on 25/May/19

\frac{3}{(x + 1) (x^{2} - 4)} = \frac{A}{x + 1} + \frac{B}{x^{2} - 4}

3 = A (x^{2} - 4) + B (x + 1)

w h e n x = - 1

3 = A (- 3)

A = - 1

w h e n x = 2

3 = B (3)

B = 1

w h e n B = - 2

3 = B (- 1)

B = - \frac{1}{3}

p a r t i a l f r a c t i o n s a r e

\frac{3}{(x + 1) (x^{2} - 4)} \equiv \frac{1}{x^{2} - 4} - \frac{1}{x + 1}

o r \frac{3}{(x + 1) (x^{2} - 4)} \equiv - \frac{1}{x + 1} - \frac{1}{3 (x^{2} - 4)}

Answered by ajfour last updated on 25/May/19

\frac{3}{(x + 1) (x + 2) (x - 2)} = \frac{a}{x + 1} + \frac{b}{x + 2} + \frac{c}{x - 2}

a = \frac{3}{(x^{2} - 4)} ∣_{x = - 1} = - 1

b = \frac{3}{(x + 1) (x - 2)} ∣_{x = - 2} = \frac{3}{4}

c = \frac{3}{(x + 1) (x + 2)} ∣_{x = 2} = \frac{1}{4}

s o

\frac{3}{(x + 1) (x^{2} - 4)} = \frac{- 1}{x + 1} + \frac{3}{4 (x + 2)} + \frac{1}{4 (x - 2)} .

Answered by malwaan last updated on 25/May/19

\frac{a}{x + 1} + \frac{b}{x + 2} + \frac{c}{x - 2}

= \frac{a (x + 2) (x - 2) + b (x + 1) (x - 2) + c (x + 1) (x + 2)}{(x + 1) (x + 2) (x - 2)}

x = - 1

\Rightarrow 3 = a (1) (- 3) = - 3 a

\Rightarrow a = - 1

x = - 2

\Rightarrow 3 = b (- 1) (- 4) = 4 b

\Rightarrow b = \frac{3}{4}

x = 2

\Rightarrow 3 = c (3) (4) \Rightarrow c = \frac{1}{4}

∴ \frac{3}{(x + 1) (x^{2} - 4)} = \frac{- 1}{x + 1} + \frac{3}{4 (x + 2)} + \frac{1}{4 (x - 2)}

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