Proving a Formula: Maximum Root Count for Polynomials

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In summary, the conversation discusses proving that a polynomial with real coefficients can have at most n roots. The approach suggested is to prove the contrapositive as being false, by assuming the polynomial has n+1 different roots and using the fact that if c is a root, then x-c is a factor of the polynomial. The conversation also mentions starting with the case where n=1 and proving that the polynomial cannot have 2 roots.
  • #1
chocolatelover
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Homework Statement


Prove that if p(x)=anx^n +an-1x^n-1+...a0, where a0,..., "an" ε reals, is a polynomial, then p can have at most n roots.


Homework Equations





The Attempt at a Solution



C ε R is a root of a polynomial p if p(c)=0. If c is a root of p, then x-c is a factor of p.

I'm not sure where to go from here. I think it would probably be the easiest to prove this by proving the contrapositive as being false. Could someone please give me a hint or show me where to go from here?

Thank you very much
 
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  • #2
Sure. Suppose the polynomial has n+1 different roots. c1,c2,...cn+1. Since c1 is a root the polynomial p(x) can be factored (x-c1)*p1(x) where p1 has degree n-1. The other c's must be roots of p1(x) since they aren't roots of (x-c1). Continue in this way until you reach degree 1. Now you have a linear polynomial with two different roots. Possible?
 
  • #3
Thank you very much

Would it be somthing like this?

(p1x)^(n-1)(x-c2)(x-c3)^(n) :confused:

Thank you
 
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  • #4
No, that's not clear at all. Start by proving if n=1 then the polynomial can't have 2 roots. Ok?
 
  • #5
Thank you very much

Regards
 

1. What is the maximum number of roots for a polynomial?

The fundamental theorem of algebra states that the maximum number of roots for a polynomial is equal to its degree. This means that a polynomial of degree 5 can have a maximum of 5 roots.

2. How can I prove the maximum root count for a polynomial?

To prove the maximum root count for a polynomial, we can use the fundamental theorem of algebra and the fact that a polynomial of degree n can be factored into n linear or complex factors. This shows that the maximum number of roots is equal to the degree of the polynomial.

3. Are there any exceptions to the maximum root count for polynomials?

Yes, there are exceptions to the maximum root count for polynomials. For example, a polynomial of degree 3 may have 3 real roots or 1 real root and 2 complex roots. In this case, the maximum number of roots is still 3, but some of the roots may be complex numbers.

4. Can a polynomial have more roots than its degree?

No, a polynomial cannot have more roots than its degree. This is because the fundamental theorem of algebra states that the maximum number of roots is equal to the degree of the polynomial.

5. How does the maximum root count for polynomials relate to its graph?

The maximum root count for a polynomial is directly related to its graph. The degree of the polynomial determines the number of times the graph will intersect the x-axis, which is equal to the number of roots. Additionally, the behavior of the graph near the x-axis can provide information about the nature of the roots (real or complex).

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