Question

Three 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.
Correct the reasoning of any inaccurate reasoning by the party-goers in full and complete sentence.Make sure you reference any theorems that support your justifications.

Answer 1

Interesting party, what with all these nerds >:)
...well, some nerds they are XD

What are your thoughts so far?
Any ideas?

Answer 2

Haha well actually I don't get how to do this I'm so lost

Answer 3

Okay, consider the Professor's theory first.

If (x+2) is a factor of a function, does it automatically mean that 2 is a zero of the function?

Answer 4

Yes?

Answer 5

It does, does it?
Consider
h(x) = (x+2)(x+3)

And let x = 2.

what is h(x)?

Answer 6

X+5x+6

Answer 7

Yes, but x = 2, and what happens?

Answer 8

You put 2 in every x so it would look like 2+10+6

Answer 9

Is that zero? :P

Answer 10

2 + 10 + 6
is that zero?

Answer 11

No

Answer 12

No. So 2 wasn't a zero of h(x)... was it? :P

Answer 13

2+10+6

Answer 14

18. Not zero.

So 2 was not a zero of h(x), since h(2) is not zero.

Understood?

Answer 15

Yes !

Answer 16

So, the professor is wrong...

Answer 17

Ok so what about ms guerra

Answer 18

Ms. Guerra gave the precise definition of a zero of a function. Can't argue with her :P

Answer 19

What about mr Romano?

Answer 20

Ever synthetic division?

Answer 21

We didn't use synthetic division

Answer 22

That's a problem.
Synthetic division with r gives you the result when a polynomial is divided by (x-r)

Answer 23

Ok well I know how to somewhat do that

Answer 24

So do I divide (x+2) by something?

Answer 25

No, all I mean is that synthetic division with 2 means it was divided by (x-2) and there was no remainder, right?

Answer 26

Yes

Answer 27

So, that means g(x) = (x-2)f(x)
for some function f(x)

Answer 28

Ok that it?

Answer 29

Well, try having x=2 now.

Answer 30

If I did this I got 1+3

Answer 31

How?

Answer 32

Ok we'll if I understood this right I plugged in my f(x) equation you showed me and I divided it by (x+2) did I do it wrong? Sorry I'm such a pain...

Answer 33

You're not a pain, lol
But in g(x) = (x-2)f(x)
Just replace all the x's with 2.

Show me what happens...step-by-step if you must ^_^

Answer 34

Ok plug in 2 in all x's I got 0 f(2)

Answer 35

0 times f(2)
0 times anything is 0
So you got g(2) equals zero after all.

Answer 36

Ooo ok now I got what you where saying

Answer 37

So, by the third nerd's reasoning, yes, it is a valid argument.
Meaning it's only (ironically) the professor who's off.

Got it? good :P

Answer 38

Than you so much for helping me I have you a medal ;)

Answer 39

No problem. Just polish it up and you'll be in shape ^_^

Answer 40

gave* sorry and um thank you again.