Proof about an odd degree polynomial

In summary, to prove that a polynomial of odd degree in R[x] with no multiple roots must have an odd number of real roots, one can refer to a Corollary stating that every polynomial of odd degree in R[x] has a root in R. This does not guarantee that all roots are in R, but rather that an odd number of them are. It is also important to consider that complex roots come in pairs, so an odd number of real roots would result from an even number of complex roots.
  • #1
chaotixmonjuish
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How could one sufficiently prove that a polynomial of odd degree in R[x] with no multiple roots must have an odd number of real roots?

My book just refers back to a Corollary that states every polynomial of odd degree in R[x] has a root in R. However it doesn't say, all roots are in R.
 
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  • #2
chaotixmonjuish said:
However it doesn't say, all roots are in R.

That's right. You're not trying to prove that all the roots are in R, merely that an odd number of them are. (You actually can't prove that all the roots are real. It's not hard to come up with a cubic equation with exactly one real root.)

One thing to ask yourself is whether complex roots have to come in pairs.
 
  • #3
Well I'm guessing its because all the complex roots will come as conjugate pairs, i.e. there are an even number of complex roots, so there will be an odd number of real roots left over.
 

1. What is an odd degree polynomial?

An odd degree polynomial is a mathematical expression consisting of a variable raised to an odd power, such as x^3 or x^5. It is called "odd degree" because the highest power of the variable is an odd number.

2. How can I prove that a polynomial has an odd degree?

To prove that a polynomial has an odd degree, you can look at the highest power of the variable in the expression. If the highest power is an odd number, then the polynomial is of odd degree.

3. What is the significance of an odd degree polynomial?

An odd degree polynomial has certain properties that make it different from polynomials of even degree. For example, an odd degree polynomial will have at least one real root, while an even degree polynomial may not have any real roots.

4. Can an odd degree polynomial have complex roots?

Yes, an odd degree polynomial can have complex roots. However, it will always have at least one real root. The number of complex roots it has will depend on the degree of the polynomial.

5. How can we use the concept of an odd degree polynomial in real-life situations?

Odd degree polynomials can be used to model various real-life situations, such as population growth, economic trends, and physical phenomena. By understanding the properties of odd degree polynomials, we can make predictions and analyze data in these situations.

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