Igive a medal to anyone to !
3 party-goers are in the corner of the ballroom having an intense argument. you walk over to settle the debate. they are discussing a function g(x). you take out your notepad and jot down their statements.
• professor mccoy: she says that 2 is a zero of g(x) because long division with (x + 2) results in a remainder of 0.
• ms. guerra: she says that 2 is a zero of g(x) because g(2) = 0.
• mr. romano: he says that 2 is a zero of g(x) because synthetic division with 2 results in a remainder of 0.

Well, the way I see it is that both Mr. Romano, and Ms. Guerra are correct but Professor McCoy is incorrect because he said (x+2) when it should be (x-2).

The factor theorem states that:
If f(a)=0, then (x-a) is a factor

The remainder theorem states that:
If (x-a) is a factor of f(x), then f(x) / (x-a)  = 0

So if 2 is indeed a zero of f(x), then a factor must be (x-2) according to the fist which supports Ms. Guerra and also if (x-2) is indeed a factor as Ms. Guerra says then we know that f(x) / (x-a) = 0 which supports Mr. Romano

Professor McCoy is wrong because he used (x + 2) when it should be (x-2). I know this because according to the factor theorem if f(a)=0, then (x-a) is a factor. And the remainder theorem says if (x-a) is a factor of f(x), then f(x)/x-a =0.