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KVPY2016MathematicsQuadratic Equation

The remainder when the polynomial 1+x²+x⁴+x⁶+ +x²² is divided by 1+x+x²+x³+ .+x¹¹ is -

Options

  1. A0
  2. B2
  3. C1+x²+x⁴+ +x¹⁰
  4. D2 (1+x²+x⁴+ +x¹⁰ )

Correct answer

D. 2 (1+x²+x⁴+ +x¹⁰ )

Step-by-step solution

array l P(x)=1+x²+x⁴+x⁶+ +x²²= (1+x² ) (1+x⁴ ) (1+x⁴+x⁸ ) (1-x⁴+x⁸ ) Q(x)=1+x+x²+x³+ +x¹¹=(1+x) (1+x² ) (1+x⁴+x⁸ ) P(x) Q(x) = (1+x⁴ ) (1-x⁴+x⁸ ) (1+x) = 1-x⁴+x⁸+x⁴-x⁸+x¹² 1+x = 1+x¹² 1+x array Remainder. When (1+x¹² ) is divided by (1+x) is =2 Now remainder . P _ x ) divided by Q ( x ) array l =2 (1+x² ) (1+x⁴+x⁸ ) =2 (1+x²+ +x¹⁰ ) array

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