Question

Prove the Rational Zeros Theorem. [Hint: Let \(\frac{p}{q},\) where \(p\) and \(q\) have no common factors except 1 and \(-1,\) be a zero of the polynomial function $$ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} $$ whose coefficients are all integers. Show that $$ a_{n} p^{n}+a_{n-1} p^{n-1} q+\cdots+a_{1} p q^{n-1}+a_{0} q^{n}=0 $$ Now, show that \(p\) must be a factor of \(a_{0}\), and that \(q\) must be a factor of \(\left.a_{n} .\right]\)

Short answer

To prove the Rational Zeros Theorem, if \( \frac{p}{q} \) is a zero, then \( p \) must be a factor of \( a_0 \) and \( q \) must be a factor of \( a_n \).

Step-by-step solution

Define the polynomial and its zero

Consider the polynomial function given by \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \] Let \( \frac{p}{q} \) be a zero of the polynomial, where \( p \) and \( q \) are integers with no common factors other than 1 or -1. This implies that \( f \left( \frac{p}{q} \right) = 0 \)

Substitute zero into the polynomial

Substitute \( \frac{p}{q} \) into the polynomial function: \[ f \left( \frac{p}{q} \right) = a_n \left( \frac{p}{q} \right)^n + a_{n-1} \left( \frac{p}{q} \right)^{n-1} + \dots + a_1 \left( \frac{p}{q} \right) + a_0 = 0 \]

Clear the fractions

To eliminate the fractions, multiply each term by \( q^n \): \[ a_n p^n + a_{n-1} p^{n-1} q + a_{n-2} p^{n-2} q^2 + \dots + a_1 p q^{n-1} + a_0 q^n = 0 \]

Analyze the equation

Since \( p \) and \( q \) have no common factors other than 1 or -1, in order for this equation to hold, \( p \) must be a factor of \( a_0 \). Every term except the last contains a factor of \( p \), so the equation can only hold if the last term, \( a_0 q^n \), is also divisible by \( p \).

Show \( p \) must be a factor of \( a_0 \)

Given the equation \[ a_n p^n + a_{n-1} p^{n-1} q + a_{n-2} p^{n-2} q^2 + \dots + a_1 p q^{n-1} + a_0 q^n = 0 \] and considering the divisibility, we conclude \( p \) must be a factor of \( a_0 \).

Show \( q \) must be a factor of \( a_n \)

Similarly, every term except the first contains a factor of \( q \), so for the equation to hold, the first term, \( a_n p^n \), must be divisible by \( q \). Thus, \( q \) must be a factor of \( a_n \).

Key concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
In this context, we focus on a single variable polynomial of the form:
\[ f(x) = a_n x^n + a_{n-1} x^{n-1}+ \text{...} + a_1 x + a_0 \]

• - The terms are ordered by the power of the variable, from the highest power to the lowest.
  
• - Each term consists of a coefficient (e.g., \(a_n, a_{n-1}\)), the variable \(x\), and a non-negative integer exponent.
  
• - The degree of the polynomial is the highest power of \(x\) with a non-zero coefficient – in this case, \(n\).
  

Polynomials are crucial in algebra and calculus since they appear in many types of equations and functions. Understanding their behavior when manipulated, such as factoring or finding zeros, is a key mathematical skill.

Factors

In mathematics, a factor is a quantity that divides another quantity without leaving a remainder. Factors are fundamental when working with polynomials.

• - For example, if we have a polynomial \(f(x)\) and a number \(c\) such that \(f(c) = 0\), then \(c\) is a root (or zero) of the polynomial, and \(x - c\) is a factor of \(f(x)\).
  
• - This means the polynomial can be expressed as \((x - c) * Q(x)\), where \(Q(x)\) is another polynomial.
  

Rational Zeros Theorem
This theorem keys in on the idea that if you have a rational root \(\frac{p}{q}\) of a polynomial equation with integer coefficients, certain conditions about its factors must hold.
Steps to Apply the Theorem:

• 1. Identify the polynomial and its highest and constant coefficient.
  
• 2. Assume \(\frac{p}{q}\) is a root, meaning it satisfies the polynomial equation.
  
• 3. Using the structure of the polynomial and properties of divisibility, establish that \(p\) must divide the constant term \(a_0\) and \(q\) must divide the leading coefficient \(a_n\).
  

Understanding factors is essential for the Rational Zeros Theorem, as it relies on divisibility rules.

Integers

Integers are the set of whole numbers, including positive, negative numbers, and zero. They do not include fractions or decimals.

• - Example integers: -3, -2, -1, 0, 1, 2, 3.
  

In the context of polynomials, integers are vital because they often serve as coefficients and possible values for roots.
Why Integers Matter in Polynomials:

• 1. Coefficients in polynomial functions: The Rational Zeros Theorem requires polynomial coefficients to be integers. This restriction allows using properties of integers, like divisibility.
  
• 2. Rational roots: When we look for rational roots \(\frac{p}{q}\) of a polynomial, \(p\) and \(q\) must be integers. Only then can the theorem's conditions about factors and divisibility be applied.
  

Understanding integers is crucial for solving polynomial equations and applying the Rational Zeros Theorem.

Divisibility

Divisibility in mathematics refers to one integer being able to divide another integer with no remainder left. This concept is central to the Rational Zeros Theorem.

• - For example, 6 is divisible by 3 because when you divide 6 by 3, you get 2, and the remainder is 0.
  

Application to Rational Zeros Theorem:

• 1. If \(\frac{p}{q}\) is a rational zero of the polynomial \(f(x)\), substituting \(\frac{p}{q}\) into the polynomial must result in zero.
  
• 2. Multiplying through by \(q^n\) clears the fraction, resulting in an integer equation.
  
• 3. By considering the factors and coefficients in the polynomial, you can show that \(p\) must be a factor of the constant term (\(a_0\)), and \(q\) must be a factor of the leading coefficient (\(a_n\)).
  

Importance of Divisibility:
Understanding divisibility helps simplify polynomial equations and ensures accurate application of the Rational Zeros Theorem. This foundational concept shows how integer properties reduce potential rational roots to a manageable set for testing.