A polynomial of degree $n$ has at most $n$ roots.

Question

I'm auditing a number theory course. I'm trying to prove the following:

A polynomial of degree $n$ has at most $n$ roots

I know there are many proofs of this on SE, however none are particularly accessible (for me, at least). I know a bit of number theory, and a bit of abstract algebra. Any good starting points or hints?

"None are particularly accessible" how come? Besides, what coefficients do those polynomials have? — Sarvesh Ravichandran Iyer, Aug 03 '20 at 06:20
That's not the fundamental theorem of algebra. That theorem says that a complex polynomial has exactly $n$ roots (if we include multiplicity). So do you need a proof of the claim you wrote, or of the FTOA? — 5xum, Aug 03 '20 at 06:21
I assume that the coefficients are drawn from a field. Argue by induction on the degree. The assertion plainly holds if the degree is 1. Assume the result for all polynomials of degree (n-1) as your induction hypothesis. Now consider a polynomial of degree n. If it has no roots there is nothing to prove. It it has a root, factor it out and use the induction hypothesis. — student, Aug 03 '20 at 06:22
Does this answer your question? How do I prove that a polynomial F[x] of degree n has at most n roots — James Moriarty, Aug 03 '20 at 06:30
You might need this https://en.m.wikipedia.org/wiki/Polynomial_remainder_theorem and the Unique Factorization Property of a polynomial ring over a field. — eduard, Aug 03 '20 at 06:33
It will be difficult for anyone to answer this if you do not explain what troubles you have understanding those other proofs. — , Aug 03 '20 at 06:47

score 3 · Answer 1 · answered Aug 03 '20 at 06:29

3

Hint: There's a factor theorem saying that if $a$ is a root, then $p(x)$ has $x-a$ as a factor.

answered Aug 03 '20 at 06:29

A polynomial of degree $n$ has at most $n$ roots.

1 Answers1