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KVPY2017MathematicsQuadratic Equation

Let r(x) be the remainder when the polynomial x¹³⁵+x¹²⁵-x¹⁸⁵+x⁵+1 is divided by x³-x . Then

Options

  1. Ar(x) is the zero polynomial
  2. Br(x) is a nonzero constant
  3. Cdegree of r(x) is one
  4. Ddegree of r(x) is two

Correct answer

C. degree of r(x) is one

Step-by-step solution

x ¹³⁵+ x ¹²⁵- x ¹¹⁵+ x ⁵+1= k ( x ³- x )+ A x ²+ Bx + C put x =0 C =1 Put x =13= A + B +1A+B=2 .........(i) put x=-1-1=A-B+1A-B=-2 .........(ii) (i) +(i i)A=0B=2

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