Lower and Upper bounds of Polynomial equations

In summary, the conversation is about the lower and upper bounds theorem for polynomial equations. The theorem states that if a<0 and P(a) is not equal to 0, and dividing P(x) by (x-a) leads to coefficients that alternate signs, then a is a lower bound of all the roots of P(x)=0. The proof for this theorem is provided and can be found in various sources online, but the person is having trouble understanding a specific part of the proof. They are requesting for someone to explain and provide further clarification on the theorem. Two sources where the proof can be found are mentioned.
  • #1
daveclifford
4
0
Recently I am studying about theorems regarding to polynomial equations and encounter the lower and upper bounds theorem. Which states that if a<0 and P(a) not equals 0, and dividing P(x) by (x-a) leads to coefficients that alternate signs, then a is a lower bound of all the roots of P(x)=0. The proof about this statement is provided but I have troubles in understanding it(I do understand about the proof of upper bound one...), I hope someone here can explain and proof about the theorem... Thanks for the help!
 
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  • #2
It would help us enormously if you could show the proof (or a reference) and indicate which part you don't understand.
 
  • #3
(Lower Bound)(Let P(x) be any polynomial with real coefficients and a positive leading coefficient.)
Let r be a root of the given polynomial. Assuming r is not equal to 0 and that the polynomial has a nonzero constant term. Since P(r)=0, we can say that P(x)=(x-r)Q(x). Substituting in x=a, we get that P(a)=(a-r)Q(a).
Since P(a),Q(a) do not equal to 0, we can divide both sides by Q(a) to get (a-r) = P(a)/Q(a).
We know that Q(a) is either positive or negative. Since a<0 and the leading terms in Q(x) has a positive coefficient, the constant term in Q(x) has the same sign as Q(a).
If we assume that neither r nor the constant term in P(x) are zero, then that guarantees that the constant term in Q(x) must be strictly positive or negative. [Since the coefficients alternate in sign, P(a) differs in sign from the constant term in Q(x). But since the constant term in Q(x) has the same sign as Q(a) we know that P(a) and Q(a) are opposite signs, which implies that (a-r)<0, which leads to a<r.]
Sentences in square brackets are the parts which I don't understand.
 
  • #4
http://onlinemathcircle.com/wp-content/uploads/2011/07/Lecture_Polynomials.pdf [Broken] (pg 4) the whole text is in this website
http://mathweb.scranton.edu/monks/courses/ProblemSolving/POLYTHEOREMS.pdf (pg 7, No. 10) another similar text
 
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  • #5
can someone help, please? :cry:
 

1. What are lower and upper bounds of polynomial equations?

Lower and upper bounds of polynomial equations refer to the smallest and largest possible values that a polynomial equation can have. These values are determined by the coefficients and degree of the polynomial.

2. How do you find the lower and upper bounds of a polynomial equation?

To find the lower and upper bounds of a polynomial equation, you can use the Rational Root Theorem or Descartes' Rule of Signs to determine the possible rational roots and number of positive and negative roots. Then, you can test these values in the polynomial equation to see which ones produce the smallest and largest values.

3. Why are lower and upper bounds important in polynomial equations?

Lower and upper bounds are important because they provide a range of values that the polynomial equation can have, which can help in solving real-world problems and determining the behavior of the equation.

4. Can a polynomial equation have more than one lower or upper bound?

Yes, a polynomial equation can have multiple lower and upper bounds. This can happen when there are multiple roots that produce the same minimum or maximum value for the equation.

5. How do lower and upper bounds affect the graph of a polynomial equation?

The lower and upper bounds can help determine the shape and behavior of the graph of a polynomial equation. They can indicate the minimum and maximum points on the graph and can also help determine if the graph will approach or intersect the x-axis at certain points.

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