Rational Roots: What are the Other Names?

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In summary, the conversation discusses the concept of rational roots and how it relates to rational zeroes in polynomials. The question is raised about whether there are other theorems that may have different names, and if this is a common occurrence in math education. The conversation also mentions the distinction between roots and zeroes, and how different definitions may be used in different math classes. The Chinese remainder theorem is mentioned as an example, with two different explanations provided.
  • #1
Plastic Photon
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I was finishing a test today when one of the questions towards the end asked to have all the rational roots listed of a polynomial. I looked at this question and I thought: "I have never heard of 'rational roots'". Though I have heard of rational zeroes, so I just assumed these two to be the same. So far, from my web search and comparison of my book they seem the same.
So I have a question to follow this (that I don't think is homework intensive, but): Are there any of other theorems which have other common names that I may encounter, does this happen often?
Maybe I did go spacecadet in class for a while and the professor might have mentioned that the two are the same/related, but...will this be a reocurring trend through an education in math, say, up to differential equations?
EDIT: this was in college algebra.
 
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  • #2
Strictly speaking, an equation has a root, while a function, such as a polynomial, has a zero. A zero of a function, f(x), is a root of the equation f(x)= 0. It is, unfortunately, a distinction that is ignored by all but anal-rententive people like me!
 
  • #3
To answer your question - some classes may have different terms for a single concept, and sometimes a syllabus will define something that you think you already know in a different way with a piossibly different meaning. Definitions rule for each class you take.

Explanations of the Chinese remainder theorem in the hands of different folks:

http://www-math.cudenver.edu/~wcherowi/courses/m5410/ctccrt.html

(This one has a proof and some "remarks" which are really limiting conditions or definitions)

http://planetmath.org/encyclopedia/ChineseRemainderTheorem.html

They don't look identical do they?
 
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1. What are the other names for rational roots?

Some other names for rational roots include rational zeros, rational solutions, and rational factors.

2. How are rational roots different from irrational roots?

Rational roots are numbers that can be expressed as a ratio of two integers (e.g. 2/3 or -5/2), while irrational roots cannot be expressed as a ratio of two integers and often involve non-repeating decimals (e.g. √2 or π).

3. Can a polynomial have both rational and irrational roots?

Yes, a polynomial can have a combination of rational and irrational roots. For example, the polynomial x^2 - 2 has both a rational root of √2 and an irrational root of -√2.

4. How do you find the rational roots of a polynomial?

To find the rational roots of a polynomial, you can use the Rational Roots Theorem, which states that the possible rational roots of a polynomial with integer coefficients can be found by taking the factors of the constant term and dividing them by the factors of the leading coefficient.

5. Why are rational roots important in mathematics?

Rational roots are important in mathematics because they can help us solve equations and find the zeros of a polynomial. They also have many real-world applications, such as in finance and engineering, where they can be used to find the break-even points or optimal solutions.

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