Question

Describe and correct the error in listing the possible rational zeros of the function. $$ f(x)=3 x^3+13 x^2-41 x+8 $$ Possible rational zeros of \(f\) : $$ \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8} $$

Short answer

The Incorrect possible rational zeros are \(\pm \frac{1}{2}\), \(\pm \frac{1}{4}\), \(\pm \frac{1}{8}\), \(\pm \frac{3}{2}\), \(\pm \frac{3}{4}\), \(\pm \frac{3}{8}\). The correct possible rational zeros according to the Rational Root Theorem are \(\pm 1\), \(\pm 2\), \(\pm 4\), \(\pm 8\), \(\pm \frac{1}{3}\), \(\pm \frac{2}{3}\), \(\pm \frac{4}{3}\), \(\pm \frac{8}{3}\).

Step-by-step solution

Apply the Rational Root Theorem

The Rational Root Theorem states that any rational root, expressed in lowest terms, has a numerator that is a factor of the constant term, and a denominator that is a factor of the leading coefficient of the polynomial. So first write down the factors of 8 (the constant term) and 3 (the leading coefficient). Factors of 8 are: \(\pm 1, \pm 2, \pm 4, \pm 8\). Factors of 3 are: \(\pm 1, \pm 3\).

Form the possible rational roots

Now form the potential rational roots by taking each combination of a factor of 8 over a factor of 3, both positive and negative: \(\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}\).

Identify and correct the error

By comparing the original list of possible rational zeros with our calculated possible zeros, we can see that the original list improperly includes values like 1/2, 1/4, 1/8, 3/2, 3/4 and 3/8. These are not valid potential zeros according to the Rational Root Theorem because they are not formed by the factor combinations of 8 and 3.

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. These functions can be as simple as linear functions like f(x) = ax + b, with a degree of 1, or as complex as higher-degree polynomials like cubic functions (f(x) = ax^3 + bx^2 + cx + d), quartic functions, and beyond.

The degree of a polynomial function is determined by the highest power of the variable that appears in the function. For instance, the polynomial function given in the exercise, f(x) = 3x^3 + 13x^2 - 41x + 8, is a cubic polynomial because the highest exponent of x is 3. The coefficients, like the 3 in front of x^3, modify the shape and location of the graph of the polynomial function.

Understanding polynomial functions is crucial for various areas of mathematics and science, including algebra, calculus, and physics. They can be used to model real-world phenomena such as the trajectory of a projectile, economic growth models, and many other applications.

Possible Rational Zeros

Identifying possible rational zeros of a polynomial function is a fundamental step in solving equations and finding the roots of the polynomial. The Rational Root Theorem is an essential tool here, which provides a structured way to list all potential rational zeros that could be solutions to the polynomial equation.

The Rational Root Theorem posits that any potential rational root of a polynomial is a fraction, where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient. For the polynomial in the exercise, the constant term is 8 and the leading coefficient is 3.

To apply the theorem effectively, you must first find all the factors of the constant term and the leading coefficient. Then, as in the solution to the exercise, create all possible fraction combinations using these factors. It is critical to simplify these fractions to their lowest terms, as the Rational Root Theorem guarantees that the roots, if rational, will be in that simplified form. By doing this, you can uncover the complete list of potential rational zeros that may solve the polynomial equation.

Factor of Polynomial

Understanding the concept of a factor of a polynomial is central to breaking down polynomials into simpler components. In essence, if a polynomial f(x) can be expressed as (x - r)g(x), then (x - r) is called a factor of f(x), and r is a root or zero of the polynomial. Factoring helps in solving polynomial equations and simplifying complex expressions.

In the context of the Rational Root Theorem, the constants and coefficients play a vital role in determining the possible rational roots, which are essentially candidates for factors of the polynomial. Once a rational root is found, it implies the existence of a factor corresponding to that root.

For instance, if through the Rational Root Theorem you find that 2 is a zero of the polynomial f(x) = 2x^3 - 6x^2 + 4x, then you know that (x - 2) is a factor of f(x). This insight is powerful for simplifying polynomials and solving complex algebraic problems, making the study of factors an indispensable part of understanding polynomial functions.