How many real roots?
a 1 − x a 1 + a 2 − x a 2 + . . . + a 2 0 1 7 − x a 2 0 1 7 = 2 0 1 5
Given the equation above, where 0 < a 1 < ⋅ ⋅ ⋅ < a 2 0 1 7 . How many real roots does it have?
The answer is 2017.
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1 solution
Same soln!
Another way:
Let's suppose that x = a + i b is a root, then we can rationalise and by equating the imaginary part on both sides we can get b = 0 .
Consider the polynomial function f(x)=a1(a2-x)...(an-x)+a2(a1-x)(a3-x)...(an-x)+.....+an(a1-x)(a2-x)...(an-1 -x).Plugging x=a1,a2,a3,...an,we observe that f changes its sign (n-1) times.f is continuous, so by IVT f has (n-1) real roots each in(ai,ai+1).Now another root will surely lie in b/w a1 & -infinity. Now for no root the polynomial g(x)=(a1-x)(a2-x)...(an-x) becomes 0.So the given equation ,i.e., the rational function g ( x ) f ( x ) has exactly n real roots. HERE n=2017(ANS)