If 27x2+32x+163x+221-x=A3x+2+B3x+22+C1-xthen AB+BC+CA=

27x2+32x+163x+221-x=A3x+2+B3x+22+C1-x27x2+32x+16=A(3x+2) (1-x)+B(1-x)+C(3x+2)2 x=1 ⇒ 75=25C ⇒C=3x2 Coeff ⇒ 27=-3A+27C ⇒ A=0x coeff ⇒ 32=A-B+12CB=36-32=4AB+BC+CA=12

If 27x2+32x+163x+221-x=A3x+2+B3x+22+C1-xthen AB+BC+CA=

27x2+32x+163x+221-x=A3x+2+B3x+22+C1-x27x2+32x+16=A(3x+2) (1-x)+B(1-x)+C(3x+2)2 x=1 ⇒ 75=25C ⇒C=3x2 Coeff ⇒ 27=-3A+27C ⇒ A=0x coeff ⇒ 32=A-B+12CB=36-32=4AB+BC+CA=12

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$If \frac{27 x^{2} + 32 x + 16}{{(3 x + 2)}^{2} (1 - x)} = \frac{A}{3 x + 2} + \frac{B}{{(3 x + 2)}^{2}} + \frac{C}{1 - x} then AB + BC + CA =$

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answer is B.

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Detailed Solution

$\begin{array}{rcl} \frac{27 x^{2} + 32 x + 16}{{(3 x + 2)}^{2} (1 - x)} & = & \frac{A}{3 x + 2} + \frac{B}{{(3 x + 2)}^{2}} + \frac{C}{1 - x} \\ 27 x^{2} + 32 x + 16 & = & A (3 x + 2) (1 - x) + B (1 - x) + C {(3 x + 2)}^{2} \\ x & = & 1 \Rightarrow 75 = 25 C \Rightarrow C = 3 \end{array}$
$\begin{array}{rcl} x^{2} Coeff & \Rightarrow & 27 = - 3 A + 27 C \Rightarrow A = 0 \\ x coeff & \Rightarrow & 32 = A - B + 12 C \\ B & = & 36 - 32 = 4 \\ AB + BC + CA & = & 12 \end{array}$