Can you please ask your teacher which function has infinitely many zeros?
That's been answered: the zero (constant) polynomial. For confirmation, see
en.wikipedia.org
Like any constant value, the value 0 can be considered as a (constant) polynomial, called the
zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is usually undefined.
The constant polynomial P(x)=0 whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The zero polynomial is the additive identity of the additive group of polynomials. The degree of the zero polynomial is...
mathworld.wolfram.com
The constant polynomial P(x)=0 whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The zero polynomial is the additive identity of the additive group of polynomials.
That last sentence is the reason we
have to consider this a polynomial!
I would not consider f(x)=0 when talking about number of zeros of polynomials.
Suppose f(x) =0 . Then f(4)=f(.56) = f(sqrt(7))= f(2/3) = ... all equal 0. Some are saying that therefore this function f has infinitely many zeros. Again, I would exclude f(x) = 0.
Whether or not we include f(x)=0, the reason a polynomial can have an infinite number of 0 is NOT because the degree of f(x) can not be infinite.
I
feel that way, too, but I can't honestly
say it.
I consider the problem to be a trick question. It's appropriate only if this has been explicitly mentioned in the class.
(In particular, when I see "infinitely many zeros", I picture a countable infinity; that isn't what this is! No wonder we were fooled.)