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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.

Verified Answer :

Hagrid

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.

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