Can Induction Prove a Polynomial of Degree n Has At Most n Roots?

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Discussion Overview

The discussion centers around the question of whether induction can be used to prove that a polynomial of degree n has at most n roots. Participants explore various approaches to this problem, including induction, synthetic division, and considerations of multiplicity and distinct roots. The scope includes theoretical aspects of polynomial roots and their properties.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests using induction on the degree of the polynomial to prove the claim, seeking confirmation on the validity of this approach.
  • Another participant encourages experimentation with the induction method, noting its common use for such proofs.
  • A different approach involves starting with one root and using synthetic division to reduce the polynomial's degree, continuing this process until reaching a constant polynomial.
  • It is noted that a polynomial of degree n can be expressed as a product of its roots, which implies that it has at most n roots when considering multiplicity and complex roots.
  • One participant argues that the theorem regarding the number of roots is pedagogically better treated as a corollary rather than a starting point for the proof.
  • Another suggestion is to first prove that a polynomial of degree n has at least one complex root, then use this to show that the polynomial can be divided down to degree n-1, leading to the conclusion about the number of roots.
  • A participant introduces an example from the ring of integers modulo 8, where a polynomial can have more than n roots, indicating that the claim may not hold universally across all rings.
  • There is a discussion about the subtleties involved in proving the claim over a field, with one participant questioning whether being an integral domain is sufficient for the proof.

Areas of Agreement / Disagreement

Participants express differing views on the validity of using induction and the implications of polynomial roots in various mathematical contexts. There is no consensus on the best approach or the generality of the claim regarding the number of roots.

Contextual Notes

The discussion highlights the complexity of proving the number of roots for polynomials, particularly in non-field contexts, and the importance of definitions and assumptions in such proofs.

jgens
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I'm trying to prove that a polynomial function of degree n has at most n roots. I was thinking that I could accomplish this by induction on the degree of the polynomial but I wanted to make sure that this would work first. If someone could let me know if this approach will work, I would appreciate it. Thanks!
 
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Why don't you give it a shot and see what happens. Induction is probably the most common way to prove this
 
One approach is going backwards. Given one root for a polynomial of degree n, you can use synthetic division to get a polynomial of degree n-1 for the rest of the roots. Continue until the polynomial is down to 0=0. You can do this exactly n times.
 
If x_1, x_2, ..., x_n are the roots of a polynomial equation, then the polynomial is a(x- x_1)(x- x_2)\cdot\cdot\cdot(x- x_n) for some number a. Multiplied out, that has degree n.

Notice that a polynomial of degree n has exactly n roots if we count "multiplicity" (if x_1= x_2 that still counts as "two roots") and if we count complex roots. Counting only distinct real roots, there still cannot be more than n roots.
 
HallsofIvy said:
If x_1, x_2, ..., x_n are the roots of a polynomial equation, then the polynomial is a(x- x_1)(x- x_2)\cdot\cdot\cdot(x- x_n) for some number a. Multiplied out, that has degree n.


This is usually treated as a theorem of what OP's about to prove. I think it is at least most pedagogical to start out without the above, and rather let it be a corollary, thus keeping the 'natural order'.
 
What you could do, then, is prove that a polynomial of degree n has at least one (complex) solution. Then show that dividing the polynomial by x-a where a is that root, gives a polynomial of degree n-1. Since a polynomial cannot have degree less than 0, there can be, at most, n roots.
 
An example of a ring where polynomials fail to have at most n roots is the ring of integers modulo 8. The polynomial f(x)=x2-1 has, in fact, 4 distinct roots: 1, 3, 5, and 7.

Proving there are at most n roots over a field is not a trivial thing. It is not a hard, but it is subtle -- easy to overlook relevant points.
 
Last edited:
Hurkyl said:
Proving there are at most n roots over a field is not a trivial thing. It is not a hard, but it is subtle -- easy to overlook relevant points.

Is not the fact that it is an integral domain sufficient?
 

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