Is p a Root of the Linear Combination af(x) + bg(x)?

[denian]

question :

if p is a common factor of the equations f(x)=0, and g(x)=0, prove that p is also a root of the equation af(x) + bg(x) where a and b are constants.


i want to know the working.
i don't think the working will be like this :

af(p) + bg(p) = a(0) + b(0) = 0

so, i need the correct working here. thanks.

 

[arcnets]

What is a 'factor of an equation'? I don't understand...

 

[AD]

Surely you must know what factorisation of polynomials is, Arcnets.

 

[arcnets]

Yeah, OK. So f(x) and g(x) share the factor (x-p), right?
$$f(x) = 0 \ (\bmod{(x-p)}) \ \wedge \ g(x) = 0 \ (\bmod{(x-p)})$$
$$\Rightarrow$$
$$af(x) + bg(x) = a\cdot0 + b\cdot0 = 0 \ (\bmod{(x-p)})$$
IOW, the original idea is correct.

LaTeX is hard...

 

[HallsofIvy]

Yes, both arcnets and I know what a "factor of a polynomial" is- but that is not normally called a "factor of an equation". Also, you didn't say that "x-p" is a factor, you said "p is a factor".

I would have interpreted that to mean that, for example, 3 is a factor of 3x²- 6x+ 3= 0 and also a factor of 3x²+ 3x- 9= 0. However, 3 is NOT root of the equation
3x²- 6x+ 3+ 3x²+ 3x- 9= 0

What you meant to say was that p was a root of both f(x)= 0 and g(x)= 0. Then, of course, af(p)+ bg(p)= a(0)+ b(0)= 0. (And, in fact, it not required that either p or x-p be factors of f and g- in fact, it is not required that f and g be polynomials.)

 

[AD]

Yes, both arcnets and I know what a "factor of a polynomial" is- but that is not normally called a "factor of an equation". Also, you didn't say that "x-p" is a factor, you said "p is a factor".

Well, I pointed out what the original poster was driving at, I didn't write the question.

You are correct with what you say, but Denian appears to be from Malaysia, maybe he's not quite au fait with English mathematical terminology.

 

[Hurkyl]

And how will he learn the English terminology if nobody ever tells him what he meant to say?

 

[denian]

well.. actually i know what's the difference btw the factor of the equation and the root of the equation.

i didnt read the question in the book correctly, and i just type everything from the book.
but, once i read the question, i quickly make an assumption that
(x-p) is a factor of the function f and g, and f(p)= g(p) = 0 without thinking any further.

i don't have much problem with the Eng terminology in math/physics/chem but a lot in Biology. this is the first year that the student of higher school are required to learn Math and Science subjects in English. so, there might be a bit problem, since we used to learn Math & Sc in Malay (for about 10 years)

anyway, thanks for the explanation

 

[HallsofIvy]

The crucial point is that saying x= p is a root of f(x)= 0 does NOT mean that (x-p) is a factor of f(x). That is only true if f(x) is a polynomial.

 

[denian]

?
an example of what you mean HallsofIvy.