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Most Important Selected Qs for JEE AdvancedMathematicsQuadratic Equation

If p(x) is a polynomial with rational cocfficient so that for all |x| 1 ; p(x)=p ( -x+ 3-3 x^2 2 ) . then (A) p(x)=a p+a(3 x-4)^3 .

Options

  1. Ap(x)=a₀+a₁ (3 x-4 x^3 )^1+a₂ (3 x-4 x^3 )^2+ . .+ (3 x-4 x^3 )^k k(x) where k(x) is a polynomial with rational
  2. Bp^ (x)=a₁ ^ (x)+2 a₂ (x) ^ (x)+ . .+k [ . (x) |^ k-1 ^ (x) k (x+| (x)|^2 k^ (x) . . where f(x) is a polynomial
  3. Ccan't be predicted
  4. Dnone of these

Correct answer

B. p^ (x)=a₁ ^ (x)+2 a₂ (x) ^ (x)+ . .+k [ . (x) |^ k-1 ^ (x) k (x+| (x)|^2 k^ (x) . . where f(x) is a polynomial

Step-by-step solution

p ( -x+ 3-3 x^2 2 ) for |x| 1 ....(1) Let x=0, p(0), p(0)=p ( 3 2 ) Which shows p(x)-p(0) is divisible by x (x- 3 2 ) Since, p(x)-p(0) has rational coefficient and 3 2 is one of the root, - 3 2 is too a root of p(x)-p(0) Thus x (x- 3 2 ) (x+ 3 2 )=x^3- 3 4 x= 4 x^3-3 x 4 is factor of p(x)-p(0)p(x)=p(0)+ (3 x-4 x^3 ) p₁(x) (2) |x| 1p(x)=p ( -x+ 3-3 x^2 2 )3 ( -x+ 3-3 x^2 2 )-4 ( -x+ 3-3 x^2 2 )=3 x-4 x^3p₁(x)=p₁ ( -x+ 3-3 x^2 2 ) aligned & p₁(x)=p₁(0)+ (3 x-4 x^3 ) p₂(x) & p(x)=p(0)+ (3 x-4 x^3 ) p₁(0) + (3 x-4 x^3

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