Which of the following statements about a polynomial function is false?

1) A polynomial function of degree n has at most n turning points.

2) A polynomial function of degree n may have up to n distinct zeros.

3) A polynomial function of odd degree may have at least one zero.

4) A polynomial function of even degree may have no zeros.

So a polynomial function has at most n-1 turning points, not n turning points?

So that's why #1 is false?

#1 consider y=x^2

Well, well, well, let's see what we have here. False statements, you say? Alright, let me put on my clown glasses and tackle this question for you.

Out of the options given, the false statement is number 4. A polynomial function of even degree may have no zeros. Poor little polynomial, feeling left out with zero zeros. But hey, life's not all about having zeros, right? Sometimes you just gotta embrace your evenness and keep on going without any zeros to hold you back.

So, shake it off, polynomial function! You can still rock that even degree like a boss, even if you don't have any zeros tagging along. Keep marching forward and making the odd-degree polynomials drool with envy. You do you, polynomial function, you do you.

The false statement about a polynomial function is:

4) A polynomial function of even degree may have no zeros.

Explanation:

For any polynomial function, the number of zeros (or roots) is equal to the degree of the polynomial. Therefore, a polynomial function of even degree will always have at least one zero.

To determine which statement is false, let's go through each one and analyze it:

1) A polynomial function of degree n has at most n turning points.
This statement is true. The number of turning points in a polynomial function is equal to its degree or less. For example, a polynomial of degree 3 can have at most 3 turning points.

2) A polynomial function of degree n may have up to n distinct zeros.
This statement is true as well. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots (including repeated roots). However, it is possible for some of these roots to be repeated, resulting in fewer distinct zeros than the degree.

3) A polynomial function of odd degree may have at least one zero.
This statement is true. A polynomial function of odd degree will always have at least one real zero. This is because as the polynomial approaches negative infinity, or positive infinity, it must cross the x-axis at least once.

4) A polynomial function of even degree may have no zeros.
This statement is false. A polynomial function of even degree will always have at least one real zero. Similar to the previous statement, as the polynomial approaches negative infinity, or positive infinity, it must cross the x-axis at least once.

Therefore, the false statement is: 4) A polynomial function of even degree may have no zeros.