(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We have this theorem:

Let [tex]f(x)\in F[x][/tex] Then f(x) has multiple roots if and only if

[tex]gcd(f(x),f'(x))=d(x)[/tex] and [tex]d(x)\geq 1[/tex]

We went BRIEFLY over the proof and we are supposed to be able to apply it on an upcoming exam.

I'm not exactly sure how it works or what I'm looking for.

2. Relevant equations

3. The attempt at a solution

To try to get a feel for it and see if it makes sense, I did this, because I know that it has multiple roots:

let [tex]f(x)=(x-1)^2=x^2-2x+1[/tex]

then[tex]f'(x)=2x-2[/tex]

so gcd(2,1)=1, therefore multiple roots?

Am I just looking at the degree of each one? Do I divide them? How does this theorem work ?

Any examples or clarification will be appreciated.

CC

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# Multiple roots of polynomials

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