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AP EAMCET202124 Aug 2021Morning ShiftMathematicsQuadratic EquationActual

Let Q(x) be a polynomial of degree n . If Q( l )=1 and Q(2 x) Q(x+1) + 56 x+7 -8=0 , then the value of ^n C₀+ ^n C₁+ + ^n C_n is equal to

Options

  1. A32
  2. B64
  3. C8
  4. D16

Correct answer

C. 8

Step-by-step solution

Q(x) be a polynomial of degree n . Q(1)=1 and Q(2 x) Q(x+1) + 56 x+7 -8=0...(i) Put x=0 in Eq. (i), we get aligned Q(0) Q(1) + 56 7 -8 & =0 Q(0) 1 +8-8 & =0 Q(0) & =0 aligned Solving Eq. (i), we get aligned & Q(2 x) Q(x+1) + 56-8 x-56 x+7 =0 & Q(2 x) Q(x+1) - 8 x x+7 =0 & Q(2 x) Q(x+1) = 8 x x+7 ...(ii) & Q(0)=0 & aligned x is a factor of Q(x) . Let Q(x)=x P(x) aligned Q(2 x) Q(x+1) & = 8 x x+7 2 x P(2 x) (x+1) P(x+1) & = 8 x x+7 P(2 x) P(x+1 & = 4(x+1) x+7 ...(ii) aligned Put x=-1 P(-2) P(0) =0 P(-2)=0 x+2 is fact

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