There are two ways to tell if a given number is a zero of a given polynomial function.
First, and probably easiest conceptually, is:
is a zero of the polynomial function if and only if
For one of your examples:
is a zero of the polynomial function if and only if .
You can do your own arithmetic to check whether 4 is or is not a zero.
The second method is by using Polynomial Long Division. The Polynomial Long Division method relies on the following theorem:
is a zero of the polynomial function if and only if is a factor of . That is to say that there exists a polynomial function such that
This leads to the process of polynomial long division. For your other example, divide by using polynomial long division. If the result is a quotient with a remainder of zero, then is a factor meaning -5 is a zero. If there is a non-zero remainder (meaning p(x) in the theorem above is a rational function rather than a polynomial function), then is NOT a factor meaning -5 is NOT a zero.
Use the procedure for polynomial long division found at either of the following websites:
