Question

Please Help!!! I give a medal to anyone to tries!!!
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.

Answer 1

Correct the reasoning of any inaccurate reasoning by the party-goers in full and complete sentences. Make sure you reference any theorems that support your justifications
I'm pretty sure Ms. Guerra is right. But does anyone understand why the others might be wrong?

Answer 2

Please? :) @jdoe0001 @e.mccormick @shamil98 @divagirl421 @KeithAfasCalcLover @wio

Answer 3

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)

Answer 4

how do you know it's (x-2) instead of (x+2)?

Answer 5

The factor theorem states that:
\[\textit{If f(a)=0, then (x-a) is a factor}\]

The remainder theorem states that:
\[\textit{If (x-a) is a factor of f(x), then }\frac{f(x)}{x-a}=0\]

Answer 6

oooo right

Answer 7

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 \(\frac{f(x)}{x-2}=0\) which supports Mr. Romano

Answer 8

I think I understand :)

Answer 9

Give me a second

Answer 10

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.

Answer 11

EXACTLY! :)