How many roots does x^3.14 have?

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In summary, the number of roots of a polynomial is equal to its degree, counting multiplicity. With rational exponents, there are only a finite number of solutions, but with irrational exponents, there are infinitely many solutions.
  • #1
Jhenrique
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If x^5 has 5 roots, if x^3 has 3 roots and if x^10 has 10 roots, so how many roots has x^3.14 ?
 
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  • #2
Jhenrique said:
If x^5 has 5 roots,

It has one root with multiplicity 5.

if x^3 has 3 roots

It has one root with multiplicity 3.

and if x^10 has 10 roots,

It has one root with multiplicity 10.

so how many roots has x^3.14 ?

One, and the root is also a branch point.
 
  • #3
pwsnafu said:
One, and the root is also a branch point.

Why one and what's a branch point?
 
  • #4
Jhenrique is referring most likely to the degree of a function and not to just specific functions.
 
  • #5
Jhenrique said:
Why one and what's a branch point?

The only solution to ##x^{3.14}=0## is ##x=0##. There is no other choice.

A branch point is a point where, on the complex plane, traveling around that point is dependent on the direction of travel. Visually, the function ##x^{3.14}## "tears" the complex plane from the point ##x=0##.

symbolipoint said:
Jhenrique is referring most likely to the degree of a function and not to just specific functions.

Even then the statement "a polynomial of degree 10 has 10 roots" is still wrong. The correct form is "a polynomial with degree 10 has 10 roots counting multiplicity", which is a sloppy way of saying "the sum of the multiplicities of the roots of a polynomial of degree 10 is equal to 10".

Also Jhenrique, the fundamental theorem of algebra is a theorem about polynomials, ##x^{3.14}## is not a polynomial. You can't expect the pattern to continue.
 
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  • #6
Let me explain at length.

Suppose we have a polynomial ##p(x)##. Suppose that the degree is ##n##. This means
##p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0##.

Now suppose that the roots of ##p(x)## are ##x_1, x_2, \ldots, x_j##, each one different. Now the "number of roots" is ##j##. The fundamental theorem does not say ##j = n##.

Each root has what we call "multiplicity", I'm going to denote as ##m_1, m_2, \ldots, m_j##.
This means when we factor
##p(x) = (x-x_1)^{m_1} (x-x_2)^{m_2} \ldots (x-x_j)^{m_j}##
The fundamental theorem states that
  1. ##j \neq 0##, and
  2. ##m_1 + m_2 + \ldots+ m_j = n##.
This is what "the number of roots counting multiplicity" means.
 
  • #7
Jhenrique said:
If x^5 has 5 roots, if x^3 has 3 roots and if x^10 has 10 roots, so how many roots has x^3.14 ?
For n a positive integer, and a non zero, [tex]x^n= a[/tex] has n distinct roots. If n is NOT an integer it has infinitely many roots.

(don't know what you mean by "x^5 has 5 roots". Equations have roots, not polynomials.)
 
  • #8
HallsofIvy said:
For n a positive integer, and a non zero, [tex]x^n= a[/tex] has n distinct roots. If n is NOT an integer it has infinitely many roots.

Are you saying that there are infinitely many complex numbers satisfying ##x^\frac{1}{2}=1##?
 
  • #9
gopher_p said:
Are you saying that there are infinitely many complex numbers satisfying ##x^\frac{1}{2}=1##?

Yep!

[tex]x = e^{4i\pi n }[/tex] for all integers n. Granted, they are all the same complex numbers.
 
  • #10
Mentallic said:
Yep!

[tex]x = e^{4i\pi n }[/tex] for all integers n. Granted, they are all the same complex numbers.
Isn't that rather like saying that there are infinitely many numbers equal to zero, since ##n - n = 0## for all integers ##n##?
 
  • #11
jbunniii said:
Isn't that rather like saying that there are infinitely many numbers equal to zero, since n−n=0 for all integers n?

I honestly don't understand it either :smile:
 
  • #12
jbunniii said:
Isn't that rather like saying that there are infinitely many numbers equal to zero, since ##n - n = 0## for all integers ##n##?

I think it's more like saying that every linear polynomial ##mx+n## with integer coefficients has infinitely many rational roots, ##\{-\frac{nk}{mk}\}_{k\in\mathbb{Z}\setminus\{0\}}##.
 
  • #13
Actually, infinitely many roots occur only when the exponent is irrational.
For example, [itex]x^{√2} = 1[/itex] has infinitely many (complex) solutions.

Indeed, let [itex]x = e^{i\pi√2n}[/itex] for any integer n.
All those solutions are distinct, because otherwise [itex]i\pi√2m = i\pi√2n + 2k\pi i[/itex] or
[itex]√2=\frac{2k}{m-n}[/itex] would be rational, which is false.

Rational exponents have finitely many solutions because then the sequence becomes periodic, when n is a multiple of the denominator.
 
  • #14
Yes, as many have pointed out, I misspoke. With rational power, [itex]x^{m/n}= a[/itex], with m/n reduced to lowest terms, has m roots. It is only if the power is irrational, [itex]x^\alpha= a[/itex], that there are an infinite number of roots,
 

FAQ: How many roots does x^3.14 have?

How do you determine the number of roots of a polynomial equation?

The number of roots of a polynomial equation can be determined by finding the degree of the equation. The degree is the highest power of the variable in the equation. The number of roots will be equal to the degree of the equation. For example, a quadratic equation (degree 2) will have 2 roots, a cubic equation (degree 3) will have 3 roots, and so on.

Can a polynomial equation have more than one root?

Yes, a polynomial equation can have more than one root. In fact, the number of roots of a polynomial equation is equal to its degree. So a quadratic equation (degree 2) will have 2 roots, a cubic equation (degree 3) will have 3 roots, and so on.

What is meant by "complex roots" of a polynomial equation?

Complex roots of a polynomial equation are solutions that involve imaginary numbers. These solutions cannot be expressed as real numbers, and they usually come in pairs. For example, the equation x^2 + 1 = 0 has two complex roots, i and -i.

Can a polynomial equation have no roots?

Yes, a polynomial equation can have no roots. This means that there are no values of the variable that will make the equation true. This often occurs when the degree of the equation is even and all the terms have the same sign.

How can you find the roots of a polynomial equation?

The roots of a polynomial equation can be found by factoring the equation, setting each factor equal to 0, and solving for the variable. Another method is to use the quadratic formula for quadratic equations or the cubic formula for cubic equations. For higher degree equations, numerical methods such as graphing or using a calculator may be necessary to approximate the roots.

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