Explanation:
a "zero" means for the given x-value the resulting y-value is 0.
the simplest way to construct this is to create a multiplication of 3 factors, where every factor turns 0 by itself for one of the given x-values :
what simplest term is 0 for x = 2 ?
x - 2
and for x = 1 ?
x - 1
and for x = 5 ?
x - 5
so, we have
f(x) = y = (x - 2)(x - 1)(x -5)
Basically that would be already a correct answer.
I suspect we need the long form of the polynomial function, and so we need to do all the multiplications :
(x - 2)(x - 1) = x² - 2x - x + 2 = x² - 3x + 2
(x² - 3x + 2)(x - 5) = x³ - 5x² - 3x² + 15x + 2x - 10 =
= x³ - 8x² + 17x - 10
so, our full polynomial function is
f(x) = y = x³ - 8x² + 17x - 10