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if f(x) and g(x) are two polynomials such that the polynomial f(x)=xf(x^3)+x^2g(x^6) is divisible by x^2+x+1

if f(x) and g(x) are two polynomials such that the polynomial f(x)=xf(x^3)+x^2g(x^6) is divisible by x^2+x+1

Grade:11

1 Answers

Vikas TU
14149 Points
5 years ago
Roots of x^2+x+1 are w and w^2 where w is the cube roots (conjugate)
thus,
f(w) = w*f(1) + w^2*g(1) = 0
w(f(1) + wg(1)) = 0
w cannot be 0
hence, 
f(1) + w*g(1) = 0 …........................(1)
 
f(w^2) = w^2f(1) + wg(1) = 0 
w(wf(1) + g(1)) = 0
since again w is not zero,
wf(1) + g(1) = 0 …...........................(2)
eliminating w from eqns. (1) and (2) we get,
 - g(1)/f(1) = – f(1)/g(1)
   g(1)^2 = f(1)^2
  g(1)^2 - f(1)^2 = 0
 [g(1) + f(1)]  [g(1) - f(1)] = 0
g(1) = f(1)  and   g(1) = – f(1)

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