Let g(x) = f(x)16. Write g(x) as g(x) = g(x)eventerms + g(x)oddterms.
g(1) = g(1)eventerms + g(1)oddterms ...(1)
g(-1) = g(-1)eventerms + g(-1)oddterms
g(-1) = g(1)eventerms - g(1)oddterms ...(2)
(1) + (2): g(1) + g(-1) = 2*g(1)eventerms
so: g(1)eventerms = (g(1) + g(-1))/2
or g(1)eventerms = (f(1)16 + f(-1)16)/2