Suppose f(x) is a quadratic polynomial with leading coefficient 1 such that f(x) + f(-x) = 2x^2 for all x and f(2016) = 0. What is the value of f(10) ?

Question

thanks

Answer 1

if f(x) = x^2+bx+c, we have

(x^2+bx+c) + (x^2-bx+c) = 2x^2
2x^2+c = 2x^2
so, c=0 and

f(x) = x^2+bx
f(2016) = 0, so
2016^2 + 2016b = 0
b = -2016

f(x) = x^2 - 2016x = x(2016-x)

f(10) = 10(2016-10) = 10(2006) = 20060