Problem 47 - Precalculus with Calculus Previews

Question

If the leading coefficient of a polynomial function \(f\) with integer coefficients is \(1,\) then what can be said about the possible real zeros of \(f\) ?

Step-by-Step Solution

Answer

Possible rational zeros are integer divisors of the constant term.

Step 1: Understanding Polynomial Function

A polynomial function with integer coefficients is given, and its leading coefficient (the coefficient of the highest degree term) is 1. We need to determine what can be said about its possible real zeros.

Step 2: Identifying Rational Root Theorem

The Rational Root Theorem states that if a polynomial function with integer coefficients has a rational root \( \frac{p}{q} \), where \( p \) and \( q \) are integers, \( p \) divides the constant term, and \( q \) divides the leading coefficient of the polynomial.

Step 3: Considering Leading Coefficient of 1

Since the leading coefficient of the polynomial is \(1\), any possible rational root would be of the form \( \frac{p}{1} = p \). Therefore, all rational roots must be integers that divide the constant term of the polynomial.

Step 4: Concluding about Real Zeros

The possible real zeros of the polynomial, which include rational zeros, are restricted to integer values that are divisors of the constant term. Thus, any possible real zero that is rational must be an integer divisor of the constant term.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It is written in the standard form:\[f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\]where:

• \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(a_n eq 0\).
• \(x\) is the variable.
• \(n\) is a non-negative integer representing the degree of the polynomial,

Polynomial functions play a crucial role in algebra and calculus because they are continuous, differentiable, and appear frequently in modeling real-world problems. Understanding their properties, such as zeros and coefficients, helps in solving equations and graphing functions effectively.

Integer Coefficients

Integer coefficients mean that each term in the polynomial has a coefficient that is a whole number. This property is significant because it directly influences the application of theorems like the Rational Root Theorem.
With integer coefficients:

• Any rational solution or root that may exist can be systematically identified.
• It allows for simplifying calculations and using divisibility rules to find factors.

In polynomials with integer coefficients, techniques such as factoring the expression become easier and more predictable. This attribute simplifies the approach to finding potential rational roots.

Real Zeros

Real zeros of a polynomial function are the roots, or solutions, of the polynomial equation that are real numbers. These are the x-values where the polynomial graph intersects the x-axis. Finding real zeros is essential because:

• They provide insight into the behavior of the polynomial function.
• Understanding real zeros helps in the factorization of polynomials.

According to the Rational Root Theorem, particularly when the leading coefficient is 1 and coefficients are integers, potential real zeros are the integers that are divisors of the constant term.
These roots can often be found using simple arithmetic calculations and testing.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In our given exercise, the leading coefficient is 1, which simplifies the application of the Rational Root Theorem considerably.
A leading coefficient of 1 means:

• For any rational root \( \frac{p}{q} \) of the polynomial, \( q = 1 \).
• This reduces rational roots to simply the integer divisors of the constant term \(a_0\).

Having a leading coefficient of 1 ensures that if the polynomial has integer roots, they must divide the constant term. This makes it easier to list potential real zeros and verify solutions.