If alpha, beta are the roots of x2+px+q=0 and also of x2n + pnxn + qn = 0 and if α/β, β/α are the roots of xn + 1 + (x+1)n = 0 , then prove that n must be an even integer.
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1 Answers
11since. alpha and beta are the roots of x2+px+q=0, therefore alpha +beta =-p ........(1)
satisfying alpha and beta in second equation and subtracting the two equations we get
alpha2n- beta 2n+pn(alphan-betan)=0
therefore
alphan+betan=-pn...(2)
satisfying alpha/ beta in its equation we get
alphan+betan+(alpha+beta)n=0
-pn+(-p)n=0
from (1) and (2) we get
-pn+(-1)npn=0
this is true only if n is an even integer.