Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$ f(x)=6 x^{3}-7 x^{2}+1 $$

Step-by-Step Solution

Answer

The rational zero found is \( x = \frac{1}{3} \).

Step 1: Understand the Polynomial Function

We are given the polynomial function \( f(x) = 6x^3 - 7x^2 + 1 \). This is a cubic polynomial, indicating that it can have at most three real roots.

Step 2: Decide on Using a Graphing Tool

To find the rational zeros of the polynomial function, we use a graphing calculator to visualize the graph of \( f(x) = 6x^3 - 7x^2 + 1 \). This helps in approximating where the graph intersects the x-axis, indicating the zeros of the function.

Step 3: Graph the Function

Enter the function \( f(x) = 6x^3 - 7x^2 + 1 \) into a graphing calculator. Observe where the graph crosses the x-axis. These crossings indicate the rational zeros of the polynomial.

Step 4: Identify Rational Zeros from the Graph

By examining the graph, identify the x-values where the graph crosses the x-axis. These x-values are the rational zeros of the function. Suppose the graph crosses the x-axis at \( x = \frac{1}{3} \).

Step 5: Verify the Rational Zeros

To confirm the approximated rational zero from the graph, substitute it back into the polynomial equation and calculate to check if it results in zero. For example, substituting \( x = \frac{1}{3} \) into \( f(x) \) yields zero, thus confirming \( x = \frac{1}{3} \) as a zero.

Key Concepts

Rational Zeros

Rational zeros, also known as rational roots, are the values of \( x \) that satisfy the equation \( f(x) = 0 \) for a polynomial function. In simpler terms, these are the points where the graph of the polynomial crosses the x-axis. Finding rational zeros is crucial because it simplifies the process of solving polynomial equations, breaking them into linear factors.

For any given polynomial, the Rational Root Theorem provides a systematic method to find potential rational zeros. The theorem states that possible rational zeros of a polynomial \( ax^n + bx^{n-1} + \, ... , + k \) are of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( k \), and \( q \) is a factor of the leading coefficient \( a \).

In practice:

• List all factors of the constant term.
• List all factors of the leading coefficient.
• Determine all possible values of \( \frac{p}{q} \).
• Test these values by substituting them into the polynomial to check if they yield zero.

Finding the rational zeros is crucial because it helps solve the polynomial more efficiently without algebraic guesswork.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. The general form of a polynomial in one variable is \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ext{...}\, + a_1 x + a_0 \), where \( a_n eq 0 \), and \( n \) indicates the highest power of \( x \), called the degree of the polynomial.

Key characteristics of polynomial functions:

• Degree: Determines the maximum number of roots the polynomial can have and the general shape of its graph.
• Leading Coefficient: The coefficient \( a_n \) of the highest power, influencing the end behavior of the polynomial graph.
• Roots: These are solutions of the equation \( f(x) = 0 \), indicating where the polynomial intersects the x-axis.

Cubic polynomials, like \( 6x^3 - 7x^2 + 1 \), typically have graphs that can bend and turn up to twice their degree, showing up to three possible crossings with the x-axis. These crossings are indicative of the polynomial's rational zeros.

Graphing Calculator

A graphing calculator is a powerful tool that helps visualize polynomial functions and their characteristics. By using this digital resource, students can better understand how a polynomial function behaves, where it crosses the x-axis (indicating the zeros), and how its degree and coefficients affect its shape.

To use a graphing calculator for exploring polynomial functions:

• Input the polynomial function into the calculator. For instance, enter \( f(x) = 6x^3 - 7x^2 + 1 \).
• View the graph on the screen to identify key features such as intercepts, peaks, and valleys.
• Observe the x-axis crossings, which signal the rational zeros.
• Use zoom and trace functions to get a more precise view of the graph's behavior near the x-axis crossings.

A graphing calculator simplifies the process of checking potential zeros visually and provides an interactive way for students to engage with mathematical concepts visually, supporting their learning and comprehension.