Problem 40 - Precalculus with Calculus Previews

Question

List, but do not test, all possible rational zeros of $$ f(x)=24 x^{3}-14 x^{2}+36 x+105. $$

Step-by-Step Solution

Answer

Possible rational zeros are combinations of factors of 105 divided by factors of 24.

Step 1: Understand the Rational Root Theorem

The Rational Root Theorem states that for a polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 \), any rational solution \( \frac{p}{q} \) of the equation is such that \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \).

Step 2: Identify Constant and Leading Coefficient

For the polynomial \( f(x) = 24x^3 - 14x^2 + 36x + 105 \), the constant term \( a_0 \) is 105, and the leading coefficient \( a_n \) is 24.

Step 3: List Factors of the Constant Term

The constant term \( 105 \) has factors: \( \pm 1, \pm 3, \pm 5, \pm 7, \pm 15, \pm 21, \pm 35, \pm 105 \).

Step 4: List Factors of the Leading Coefficient

The leading coefficient \( 24 \) has factors: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \).

Step 5: Form All Possible Rational Zeros

Using the Rational Root Theorem, form possible rational zeros as \( \frac{p}{q} \), where \( p \) is any factor of 105 and \( q \) is any factor of 24. The set of all possible rational zeros includes fractions like \( \pm \frac{1}{1}, \pm \frac{3}{2}, \pm \frac{5}{3}, ..., \pm \frac{35}{12}, \pm \frac{105}{24} \). List down each possibility.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of the variable with constant coefficients. In simpler terms, they are equations that feature variables raised to whole number powers and do not contain division by variables. Common forms of polynomial functions include linear, quadratic, cubic, and higher-degree functions. They can be expressed as: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where each \( a_i \) represents a coefficient and \( n \) is the degree of the polynomial, which tells you the highest power of \( x \) in the polynomial.

• The degree of a polynomial determines its general shape and behavior as the variable value increases or decreases.
• Polynomial functions are smooth and continuous, meaning they have no breaks, holes, or sharp corners.

Understanding polynomial functions is crucial because they encompass many real-world phenomena, allowing us to describe everything from motion to finance. When you're dealing with functions like \( f(x) = 24x^3 - 14x^2 + 36x + 105 \), knowing the degree helps to predict its behavior and potential intersections with the x-axis.

Rational Zeros

The Rational Root Theorem is a fundamental tool to find potential rational zeros (roots) of a polynomial. A rational number is any number that can be expressed as the fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q eq 0 \). Using this theorem, we can identify possible rational roots for a polynomial function by examining the factors of the constant term and the leading coefficient.

• The constant term \( a_0 \) provides the list of potential numerators (\( p \)).
• The leading coefficient \( a_n \) provides the list of potential denominators (\( q \)).

This means that if a polynomial has any rational zeros, they must be in the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Though not all combinations will be zeros of the polynomial, eliminating non-viable options leads to efficient identification of actual rational roots.

Factors of Coefficients

Finding rational zeros using the Rational Root Theorem involves understanding the factors of the constant term and the leading coefficient of the polynomial. To illustrate:

• Let's consider the polynomial \( f(x) = 24x^3 - 14x^2 + 36x + 105 \).
• The constant term \( a_0 = 105 \) has factors \( \pm 1, \pm 3, \pm 5, \pm 7, \pm 15, \pm 21, \pm 35, \pm 105 \).
• The leading coefficient \( a_n = 24 \) has factors \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \).

By taking each factor of the constant \( a_0 \) and dividing it by each factor of the leading coefficient \( a_n \), we create a comprehensive list of potential rational zeros. These derived fractions include possibilities like \( \pm \frac{1}{1}, \pm \frac{3}{2}, \pm \frac{5}{3} \), and so on. Understanding these factors helps narrow down potential answers, making it easier to verify actual zeros through testing or substitution in the polynomial equation.