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.

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.

Answer 2

@phi

Answer 3

let's make up a simple polynomial that has a zero at x=2
for example: g(x)= x-2
when does g(x)=0 ?
when x- 2 = 0
or x= 2

now test the first statement
long division with (x + 2) results in a remainder of 0.

(x-2)/(x+2) what do you get ?

Answer 4

x^2+2x-2x-4
x^2-4

Answer 5

when you divide, you get something smaller. But the important point here is it does not go evenly. statement one is wrong.

Answer 6

statement 2 is the definition of a zero, so it is correct.

to test the last statement, try synthetic division
2 | 1 -2
|____2__
1 0

Answer 7

Oh, I did FOIL. I'm not sure how to divide them

Answer 8

when you have time, here is how to do it
http://www.khanacademy.org/math/algebra/multiplying-factoring-expression/dividing_polynomials/v/polynomial-division

but it does not divide evenly

btw, in the same play list is a video on synthetic division.

Answer 9

I've done synthetic dvision

Answer 10

What about Ms. Guerra? She says that 2 is a zero of g(x) because g(2) = 0.

Answer 11

@phi ?

Answer 12

I told you up above.

statement 2 is the definition of a zero, so it is correct.

Answer 13

Oh I'm sorry. I missed that post. lol

Answer 14

So is the last statement correct since it equals zero?

2 | 1 -2
|____2__
1 0

Answer 15

@phi ...

Answer 16

Are their steps I need to take after that?