Proving theorem for polynomials

In summary, the conversation discusses a proof of the statement that for two polynomials with the same degree, all coefficients must be equal for the polynomials to be equal. The solution provided uses the method of contradiction and the concept of roots to prove the statement. The contradiction arises from the fact that if there exists an index where the coefficients are not equal, the polynomial would have more roots than allowed by Theorem 2, leading to a contradiction.
  • #1
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Homework Statement


Prove the following statement:

Let f be a polynomial, which can be written in the form
fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0
and also in the form
fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0

Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n

Homework Equations


3. The Attempt at a Solution [/B]
0 = f(x) - f(x) = (a(n)-b(n))X^(n) + (a(n)-b(n))X^(n) + ... + a0 - b0
Need to prove that Di = (a(i)-b(i)) = 0 for i = 0,1,2,3,4,...n

But I do not know how.

This is the answer from book, using the method of contradiction:
Suppose that there exists some index i such that Di does not equal 0.
Let m be the largest of these indices, so that we can write
0 = D(m)X^(m) + ... + D0 for all x and D(m) does not equal 0. This contradicts Theorem 2. Therefore we conclude that Di = 0 for all i = 1, . . . , n, thus proving the theorem.

Theorem 2 is as stated:
Let f be a polynomial. Let a0,a1,...,a(n-1),a(n) be numbers such that a(n) does not equal 0, and such that we have: f(x) = a0 + a1X + ... + a(n)X^(N) for all x. Then f has at most n roots.

I don't understand the contradiction between theorem 2 and the statement regarding D(m). Could someone explain?
 
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  • #2
The polynomial with the D coefficients is zero for all x and therefore every real number is a root of it. Therefore it has more than m roots. But theorem 2 says it cannot have more than m roots.
 
  • #3
Thank you, answers it perfectly
 

1. What is the process for proving a theorem for polynomials?

The process for proving a theorem for polynomials typically involves breaking down the theorem into smaller, more manageable steps. These steps often include simplifying expressions, applying algebraic rules and properties, and using mathematical induction. It is also important to clearly state and justify each step in the proof.

2. Can a polynomial theorem be proven using different methods?

Yes, there are often multiple ways to prove a polynomial theorem. Some common methods include direct proof, proof by contradiction, and proof by induction. The method used will depend on the specific theorem and the preferences of the mathematician.

3. How important is it to understand the properties of polynomials before attempting to prove a theorem?

It is crucial to have a strong understanding of polynomial properties in order to successfully prove a theorem. This includes knowing the definitions of terms such as degree, leading coefficient, and roots, as well as understanding how to perform operations on polynomials.

4. Can a polynomial theorem be proven using examples?

While examples can be helpful in understanding a polynomial theorem, they are not enough to prove the theorem. A proof requires logical and mathematical reasoning, rather than just demonstrating the theorem with a few specific examples.

5. Are there any common mistakes to avoid when proving a theorem for polynomials?

One common mistake when proving a polynomial theorem is assuming that a particular property or rule applies without providing justification. It is important to clearly explain each step in the proof and show how it follows logically from the previous steps. Additionally, it is important to check for any errors or mistakes in calculations, as even a small error can invalidate the entire proof.

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