express-7x-4-x-3-x-2-9x-9-in-partial-fraction-

Question Number 35245 by JOHNMASANJA last updated on 17/May/18

e x p r e s s \frac{7 x + 4}{x^{3} + x^{2} + 9 x + 9} i n p a r t i a l

f r a c t i o n

Commented by Rasheed.Sindhi last updated on 17/May/18

prof Abdo imad

Use of limit in partial fractions is new

for me . Could you please insert more

steps to discribe 4 th, 5 th & 6 th lines ?

Commented by Rasheed.Sindhi last updated on 17/May/18

Sir th \propto n k $ a ⌊ ♡ ⊤!

BTW

prof Abdo imad \overset{?}{=} abdo mathsup 649 cc

\overset{?}{=} math khazana by abdo

\overset{?}{=} abdo . msup . com

\overset{?}{=} ID containing^{'} {abdo}^{'}

? ? ?

Commented by prof Abdo imad last updated on 17/May/18

l e t p u t F (x) = \frac{7 x + 4}{x^{3} + x^{2} + 9 x + 9}

F (x) = \frac{7 x + 4}{x^{2} (x + 1) + 9 (x + 1)} = \frac{7 x + 4}{(x + 1) (x^{2} + 9)}

F (x) = \frac{a}{x + 1} + \frac{b x + c}{x^{2} + 9}

a = {l i m}_{x \to - 1} (x + 1) F (x) = \frac{- 3}{10}

{l i m}_{x \to + \infty} x F (x) = 0 = a + b \Rightarrow b = \frac{3}{10} \Rightarrow

F (x) = \frac{- 3}{10 (x + 1)} + \frac{\frac{3}{10} x + c}{x^{2} + 9}

F (0) = \frac{4}{9} = \frac{- 3}{10} + \frac{c}{9} \Rightarrow 4 = - \frac{27}{10} + c \Rightarrow

c = 4 + \frac{27}{10} = \frac{67}{10} \Rightarrow

F (x) = \frac{- 3}{10 (x + 1)} + \frac{1}{10} \frac{3 x + 67}{x^{2} + 9} .

Commented by Rasheed.Sindhi last updated on 17/May/18

x^{3} + x^{2} + 9 x + 9 = x^{2} (x + 1) + 9 (x + 1)

= (x + 1) (x^{2} + 9)

\frac{7 x + 4}{(x + 1) (x^{2} + 9)} = \frac{A}{x + 1} + \frac{B x + C}{x^{2} + 9}

7 x + 4 = A (x^{2} + 9) + (B x + C) (x + 1)

x = - 1 \Rightarrow - 3 = 10 A \Rightarrow A = - \frac{3}{10}

x = \pm 3 i \Rightarrow 4 \pm 21 i = (B (\pm 3 i) + C) (\pm 3 i + 1)

4 \pm 21 i = - 9 B \pm 3 i B \pm 3 i C + C

= - 9 B + C \pm 3 i (B + C)

- 9 B + C = 4 \land B + C = 7

\Rightarrow B = \frac{3}{10} \land C = \frac{67}{10}

\frac{7 x + 4}{(x + 1) (x^{2} + 9)} = \frac{- 3 / 10}{x + 1} + \frac{(3 / 10) x + (67 / 10)}{x^{2} + 9}

= - \frac{3}{10 (x + 1)} + \frac{3 x + 67}{10 (x^{2} + 9)}

Commented by Rasheed.Sindhi last updated on 18/May/18

⊤ ∦ \forall \cap ∣ ⟨ $!

Commented by abdo mathsup 649 cc last updated on 17/May/18

s i r R e s h e e d t h i s m e t h o d i s g e n e r a l a n d e a s y

f o r a w e h a v e F (x) = \frac{a}{x + 1} + \frac{b x + c}{x^{2} + 9} \Rightarrow

{l i m}_{x \to - 1} (x + 1) F (x) = a + {l i m}_{x \to - 1} (x + 1) \frac{b x + c}{x^{2} + 9}

= a f r o m a n o t h e r s i d e

{l i m}_{x \to - 1} (x + 1) F (x) = {l i m}_{x \to - 1} \frac{7 x + 4}{x^{2} + 9} = \frac{- 3}{10} \dots .

w e d o t h e s a m e f o r b a n d c \dots

Commented by abdo mathsup 649 cc last updated on 17/May/18

y e s y e s a l l t h i s p r o f i l s a r e f o r m e b u t n o t a l l I D

c o n t a i n i n g a b d o \dots .