MHB Polynomial Challenge: Find Real Solutions

[anemone]

Find the number of distinct real solutions of the equation

$(x − 1)(x − 3)(x − 5) · · · (x − 2017) = (x − 2)(x − 4)(x − 6) · · · (x − 2016)$.

 

[kaliprasad]

Find the number of distinct real solutions of the equation

$(x − 1)(x − 3)(x − 5) · · · (x − 2017) = (x − 2)(x − 4)(x − 6) · · · (x − 2016)$.

Spoiler

the above is same as
$P(x) = (x - 1)(x - 3)(x - 5) \cdots (x - 2017) - (x - 2)(x - 4)(x - 6) \cdots (x - 2016)= 0$
This is a polynomial of degree 1009.
this has product of 1009 terms in the 1st term
now let us compute P(2n) for n = 1 to 1008
this is +ve for n odd( as in 1st term there are n positive and 1009-n -ve terms
and 2nd term is zero) and -ve for n even. and $P(-\infty) < 0$ and $P(\infty) > 0$ so sign changes 1009 times
so 1009 real roots.