x   f(x)
____________
-5   0
-1   36
0   0
1   6
2   0

The function f is defined by a polynomial. Some values of x and f(x) are shown in the table above. Which of the following could define f?

A) (x – 5) (x + 2)
B) (x + 5)^2 (x – 2)^3
C) x^2 (x + 5) (x – 2)^2
D) x (x + 5) (x – 2)

Specifically, can you explain why C is correct but D cannot be?

OK, so first note that because the table shows you , , and , you know the polynomial must have factors of , (AKA plain old ), and .

I sense you got this far because you’re specifically looking for the difference between choice C and choice D. (Choices A and B don’t have as a factor.)

But both choices C and D have all the factors we need, so how do we eliminate one?

Well, we plug in. Use the easier expression, which is choice D, and pick the easiest number from the table to plug in, which is 1. When you plug in 1 for , the table says you should get 6. Welp, that didn’t work! Guess we can cross off D. Just for kicks, let’s make sure choice C does work:  Robin says: