Real Roots of Polynomial Equations: Proving Equality of Real Roots

In summary, by the Fundamental Theorem of Calculus, a polynomial of odd degree with real coefficients will have at least one real root. This also applies to the polynomial P(P(x))=0. Additionally, by the FTC, both P(x) and P(P(x)) factor into complex linear factors. By setting \alpha_1,...,\alpha_m as the roots of P(x)=0 and finding distinct \beta_1,...,\beta_n such that P(\beta_i) = \alpha_i, we can show that the equation P(P(x))=0 has at least as many real roots as the equation P(x) = 0, counted without multiplicities.
  • #1
2,020
1
[SOLVED] roots of a polynomial

Homework Statement


Let P(x) be a polynomial of odd degree with real coefficients. Show that the equation P(P(x))=0 has at least as many real roots as the equation P(x) = 0, counted without multiplicities.

Homework Equations


By the FTC, P(x) and P(P(x)) factor into complex linear factors.

The Attempt at a Solution


Please just give me hint.

By the odd degree, we know that both P(x) and P(P(x)) have at least one real root.

By the FTC, P(x) and P(P(x)) factor into complex linear factors.

Oh wait, let \alpha_1,...,\alpha_m be the roots of P(x)=0. Because P(x) has odd degree, we know that p(R) = R. So, we can find distinct \beta_1,...,\beta_n such that P(\beta_i) = \alpha_i. That was easy. I guess I will post it anyway.
 
Physics news on Phys.org
  • #2
ehrenfest said:
I guess I will post it anyway.

:uhh:
 
  • #3
cristo said:
:uhh:

OK fine delete it.
 

What are the roots of a polynomial?

The roots of a polynomial are the values of the independent variable that make the polynomial equation equal to zero.

How do you find the roots of a polynomial?

To find the roots of a polynomial, you can use various methods such as factoring, the quadratic formula, or synthetic division.

Why are the roots of a polynomial important?

The roots of a polynomial are important because they help us find the x-intercepts of the polynomial graph, which can provide valuable information about the behavior of the function.

Can a polynomial have more than two roots?

Yes, a polynomial can have multiple roots. The number of roots of a polynomial is equal to its degree, but some roots may have multiplicity, meaning they appear more than once.

What is the relationship between the roots of a polynomial and its factors?

The roots of a polynomial are the solutions to the equation, while the factors are the expressions that make up the polynomial. The roots and factors are related through the polynomial's fundamental theorem, which states that the roots are the same as the factors of the polynomial.

Suggested for: Real Roots of Polynomial Equations: Proving Equality of Real Roots

Back
Top