Polynomial Challenge: Show Real Roots >1 Exist

In summary, a polynomial is an algebraic expression consisting of variables and coefficients that can be used to represent mathematical relationships. "Show real roots >1 exist" refers to finding solutions to a polynomial equation with values greater than 1. This can be proven using various mathematical techniques such as the Rational Root Theorem or Descartes' Rule of Signs. It is important to show that real roots >1 exist for a polynomial as it provides evidence of at least one solution that satisfies given conditions, which can be useful in various applications. A polynomial cannot have an infinite number of real roots >1 as the degree of the polynomial determines the maximum number of real roots it can have.
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anemone
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If the equation $ax^2+(c-b)x+e-d=0$ has real roots greater than 1, show that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.
 
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Suppose $p(x)=ax^4+bx^3+cx^2+dx+e=0$ has no real root.

Let $y>1$ be a root of $ay^2+(c-b)y+e-d=0$ and $z=\sqrt{y}$.

Since $p(x)=ax^4+(c-b)x^2+(e-d)+(x-1)(bx^2+d)$, we get

$p(z)=(z-1)(bz^2+d)$ and $p(-z)=(-z-1)(bz^2+d)$.

Now, $z>1$ implies one of $p(z)$ and $p(-z)$ is positive while the other is negative. Therefore, $p(x)$ has a root between $z$ and $-z$, a contradiction.
 

FAQ: Polynomial Challenge: Show Real Roots >1 Exist

What is a polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. It can have one or more terms, and the highest power of the variable is called the degree of the polynomial.

What is the challenge of showing real roots >1 exist for a polynomial?

This challenge involves proving that a polynomial equation has at least one real root with a value greater than 1. In other words, it requires finding a value of x that satisfies the equation and is greater than 1.

Why is it important to show real roots >1 exist for a polynomial?

Proving the existence of real roots greater than 1 for a polynomial is important in many applications, such as optimization problems, engineering, and physics. It helps in determining the behavior and solutions of a system or equation.

What methods can be used to show real roots >1 exist for a polynomial?

There are various methods that can be used to show the existence of real roots >1 for a polynomial, such as the Intermediate Value Theorem, Descartes' Rule of Signs, and the Rational Root Theorem. These methods involve analyzing the coefficients and properties of the polynomial to determine the existence of real roots.

Are there any polynomials for which it is impossible to show real roots >1 exist?

Yes, there are some polynomials for which it is impossible to show the existence of real roots >1. For example, a polynomial with all complex roots or a polynomial with no real roots. In such cases, it is not possible to find a value of x that satisfies the equation and is greater than 1.

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