lf 3x+2x+12x2+3=Ax+1+Bx+C2x2+3, then A+C-B is equal to

3x+2(x+1) (2x2+3)=Ax+1+Bx+c2x2+3 3x+2=A(2x2+3)+(Bx+C) (x+1) x=-1 ⇒ -1=A(5) ⇒ A=-15 x2 Coeff ⇒ 0=2A+B ⇒ B=25 Const ⇒ 2=3A+C ⇒ C=135 A-B+C=-15-25+135 =105=2

lf 3x+2x+12x2+3=Ax+1+Bx+C2x2+3, then A+C-B is equal to

3x+2(x+1) (2x2+3)=Ax+1+Bx+c2x2+3 3x+2=A(2x2+3)+(Bx+C) (x+1) x=-1 ⇒ -1=A(5) ⇒ A=-15 x2 Coeff ⇒ 0=2A+B ⇒ B=25 Const ⇒ 2=3A+C ⇒ C=135 A-B+C=-15-25+135 =105=2

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$lf \frac{3 x + 2}{(x + 1) (2 x^{2} + 3)} = \frac{A}{x + 1} + \frac{Bx + C}{2 x^{2} + 3}, then A + C - B is equal to$

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Detailed Solution

$\frac{3 x + 2}{(x + 1) (2 x^{2} + 3)} = \frac{A}{x + 1} + \frac{Bx + c}{2 x^{2} + 3} 3 x + 2 = A (2 x^{2} + 3) + (Bx + C) (x + 1) x = - 1 \Rightarrow - 1 = A (5) \Rightarrow A = \frac{- 1}{5} x^{2} Coeff \Rightarrow 0 = 2 A + B \Rightarrow B = \frac{2}{5} Const \Rightarrow 2 = 3 A + C \Rightarrow C = \frac{13}{5} A - B + C = \frac{- 1}{5} - \frac{2}{5} + \frac{13}{5} = \frac{10}{5} = 2$