Question

Let p(x) be a unique polynomial of minimal degree with the following properties. 1. p(x) has a leading coefficient of one. 2. 1 is a root of p(x)-1. 3. 2 is a root of p(x-2). 4. 3 is a root of p(3x). 5. 4 is a root of 4p(x). The roots of p(x) are integers with one key exception. The root that is not an integer can be written in the form of m/n, where m and n are relatively prime positive integers. What is m/n?

Answer 1

The polynomial p(x) with the given properties can be written as (x-1)(x-2+2)(x-9/3)(x-4/4). The root that is not an integer is -1/3.

Step-by-step explanation:

A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division

The polynomial p(x) with the given properties can be written as follows:

p(x) = (x-1)(x-2+2)(x-9/3)(x-4/4)

Therefore, p(x) = (x-1)(x)(3x-9)(x-1)

To find the value of m/n, we need to determine the root that is not an integer. From the equation, we can see that the root -1/3 is not an integer. Therefore, m/n = -1/3.