We can use the zeros of a polynomial function to figure out what an expression for the polynomial might be. One way to write a polynomial expression is as a product of linear factors. 

For example, for a polynomial function that satisfies when is -1, 2, or 4, we could multiply together a factor that is 0 when , a factor that is 0 when , and a factor that is 0 when . It turns out that there are many possible expressions for .

Using linear factors, one possibility is .

Another possibility is , since the 2 (or any other rational number) does not change what values of make the function equal to 0.

We can test the three values -1, 2, and 4 to make sure that is 0 for those values. We can also graph both possible versions of  and see that the graphs intercept the horizontal axis at -1, 2, and 4. Notice that while both functions have the same output at these three specific input values, they have different outputs for all other input values. 

Graph of 2 polynomial functions, one solid and one dashed, xy-plane, origin O.  Horizontal axis, scale -5 to 4 by 1’s. Vertical axis, scale -20 to 18, by 2’s. Both functions start as increasing in Quadrant 3 and end increasing in Quadrant 1. Both functions have the same roots that are increasing through (-1 comma 0), decreasing through (2 comma 0) and increasing through (4 comma 0). The solid function has a local maximum at (0 comma 8) and a local minimum at (3 comma -4). The dashed function has a local maximum at (0 comma 16) and a local minimum at (3 comma -8).