Proving Continuity of a Polynomial Function at a Zero of Another Polynomial

[kathrynag]

Homework Statement

Let p and q be a polynomial and x₀ be a zero of q of multiplicity m. Prove that p/q can be assigned a value at x₀ such that the function thus defined will be continuous there iff x₀ is a zero of p of multiplicity greater than or equal to m.

Homework Equations

The Attempt at a Solution

I'm not quite sure how to even get started. This question just confuses me in what it's asking.

 

[Office_Shredder]

If f(x) is a polynomial, and has a zero at x₀ of degree m, then there exists a polynomial g(x) such that f(x) = g(x)*(x-x₀)^m

As written, p/q (if q has a zero at x₀) is not defined at x₀, but is defined continuously in a ball around x₀. A standard trick is to extend functions by finding what their limit is as you approach points that aren't in the original domain, and then defining a new function that equals the original function on the original domain, and if you're on one of the non-defined points, you define it to be the limit of the original function.

So try writing p and q in the form I gave at the top of my post, and then use the tip to consider the relative degrees of the roots and figure out if the limit as x approaches x₀ exists

 

[kathrynag]

If f(x) is a polynomial, and has a zero at x₀ of degree m, then there exists a polynomial g(x) such that f(x) = g(x)*(x-x₀)^m

As written, p/q (if q has a zero at x₀) is not defined at x₀, but is defined continuously in a ball around x₀. A standard trick is to extend functions by finding what their limit is as you approach points that aren't in the original domain, and then defining a new function that equals the original function on the original domain, and if you're on one of the non-defined points, you define it to be the limit of the original function.

So try writing p and q in the form I gave at the top of my post, and then use the tip to consider the relative degrees of the roots and figure out if the limit as x approaches x₀ exists

Ok so p: there exists a polynomial q such that p=q*(x-x₀)^m
Is this right then do the same for q?

 

[kathrynag]

Ok so p: there exists a polynomial q such that p=q*(x-x₀)^m
Is this right then do the same for q?

or p=r*(x-x₀)^m?

 

[HallsofIvy]

Office Shredder said p(x)= g(x)(x- x₀)^m, NOT "q(x)(x- x₀)^m". You cannot use "q" to mean two different polynomials.

Don't write "p(x)= q(x)(x- x₀)^m" or "p= r(x- x₀)^m). Do what Office Shredder suggested!

 

[kathrynag]

Ok, so I have p=g(x)(x-x_{0})^{m}
q=g(x)(x-x_{0})^{n}
p/q=(x-x_{0})^{m-n}

 

[kathrynag]

I'm just confused on where to go next... Like do I do a delta, epsilon proof?