# Decidability of Polynomials with Integer Coefficients and at Least 1 Real Root?

• Dragonfall
In summary, the question is asking for a proof that the set of polynomials with integer coefficients and at least one real root is decidable. The suggested approach is to use a finite algorithm to determine if an integer coefficient polynomial has a real root, potentially by using the zeros of its derivative and reducing it to a recursive problem. However, it is unclear if this is the correct approach.
Dragonfall

## Homework Statement

Show that the set of polynomials with integer coefficients with at least 1 real root is decidable.

## The Attempt at a Solution

The question did not ask for specific language, just an intuitive finite algorithm will do.

In other words, how do you determine whether an integer coefficient polynomial in one variable has at least one real root?

Anyone?

I was thinking maybe by finding the zeros of the derivative, etc, and thus reducing the problem to a recursive one, but I don't know how to do this precisely, or know whether this is the right approach at all.

## 1. What is a polynomial?

A polynomial is a mathematical expression that consists of variables, coefficients, and exponents. It can have one or more terms and can be added, subtracted, multiplied, and divided.

## 2. What are the different types of polynomials?

The different types of polynomials include monomials, binomials, trinomials, and higher degree polynomials. Monomials have only one term, binomials have two terms, and trinomials have three terms.

## 3. How do you simplify a polynomial?

To simplify a polynomial, you need to combine like terms by adding or subtracting their coefficients. You can also use the distributive property to remove parenthesis and combine terms.

## 4. What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of the variable. For example, in the polynomial 3x^5 + 2x^3 + 4, the degree is 5 because it is the highest exponent.

## 5. How do you solve a polynomial equation?

To solve a polynomial equation, you need to set the polynomial equal to zero and use algebraic methods such as factoring, the quadratic formula, or synthetic division to find the roots or solutions of the equation.

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