pls answer this question . pls dont post replies like - i dont know
Answers
alpha = a, beta = b
given,
a and b are roots of x^2 + px + q = 0
means x = a or x = b
also
a + b = -p
ab = q
again a and b are roots of x^2n + p^n x^n + q^n = 0
means x = a or x = b
so x^n = a^n or x^n = b^n
now,
x^2n + p^n x^n + q^n = 0
(x^n)^2 + p^n x^n + q^n = 0
this is a quadratic equation whose roots are
x^n = a^n or x^n = b^n
so
sum of roots = -p^n
a^n + b^n = - p^n
a^n b^n = q^n
now
a/b , b/a are roots of
(x + 1)^n + x^n + 1 = 0
putting x = a/b we get
(a/b + 1)^n + (a/b)^n + 1 = 0
(a + b)^n / b^n + a^n / b^n + 1 = 0
(a + b)^n + a^n + b^n = 0
( if we keep x = b/a in the equation, we will get the same expression as above)
so in both cases
(a + b)^n + a^n + b^n = 0
now a+ b = -p
a^n + b^n = -p^n
thus
( - p )^n + (- p^n) = 0
( - p )^n - p^n = 0
since we have put the roots in the equation, the value must be zero..
since p is not zero, for the value to be equal to zero
( - p )^n should be equal to p^n
this is possible only when n is even