Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Modeling Polynomials A third-degree polynomial function \(f\) has real zeros \(-1,2\), and \(\frac{10}{3} .\) Find two different polynomial functions, one with a positive leading coefficient and one with a negative leading coefficient, that could be \(f\). How many different polynomial functions are possible for \(f ?\)

Step-by-Step Solution

Answer

Two possible polynomial functions could be: \(f(x) = (x+ 1)(x - 2)(x - \frac{10}{3})\) and \(f(x) = -1(x+ 1)(x - 2)(x - \frac{10}{3})\). The number of possible different polynomial functions for f is 2, considering only positive and negative leading coefficients.

Step 1: Formulate the Basic Polynomial Function

Knowing that a polynomial function \(f\) of degree 3 has real zeros \(x = -1, 2, \frac{10}{3}\), we can write the basic polynomial function as \(f(x) = k(x + 1)(x - 2)(x - \frac{10}{3})\) where \( k \) is a constant (i.e., the leading coefficient).

Step 2: Determine the Two Functions from the Basic Polynomial

Now, let's determine two examples of such functions having positive and negative leading coefficients respectively. We can do this by choosing a positive and a negative number for \( k \). \ 1. For a positive leading coefficient, let \( k = 1\). Substituting this into the initial equation, we obtain \(f(x) = (x+ 1)(x - 2)(x - \frac{10}{3})\). 2. For a negative leading coefficient, let \( k = -1\). Using this value for \( k \) in the equation, we get \(f(x) = -1(x+ 1)(x - 2)(x - \frac{10}{3})\).

Step 3: Determine the Number of Possible Functions for f

To find the number of possible polynomial functions for \( f \), remember that any non-zero real number for \( k \) would form a valid polynomial function. This is because a nonzero constant multiple of a polynomial function does not change its zeros. Hence, there are infinitely many polynomial functions that we could define to satisfy this condition. But, since we are only considering whether the leading coefficient is positive or negative, there are 2 possible polynomial functions for \( f \).

Key Concepts

Third-Degree Polynomials

A third-degree polynomial, also known as a cubic polynomial, is a polynomial of the form \( ax^3 + bx^2 + cx + d \). The degree of the polynomial is determined by the highest power of \( x \) present, which in this case is 3.

Third-degree polynomials can include:

• Variable terms raised to the third power.
• Various constants and coefficients.
• Up to three real zeros or roots where the function equals zero.

The structure of these polynomials makes them quite flexible for modeling real-world phenomena ranging from physics to finance.

These polynomials can be graphically represented, typically resulting in a curve that will always intersect the y-axis and could have one or two bends, depending on its equation's roots and coefficients.

Real Zeros

Real zeros of a polynomial function are the x-values where the function evaluates to zero. For a third-degree polynomial, there can be up to three real zeros, depending on whether they are distinct or repeated.

Identifying these zeros gives us critical insights into the behavior of a polynomial's graph:

• Each zero is a point where the graph crosses the x-axis, indicating a change in sign.
• The multiplicity of a zero indicates how the graph interacts with the x-axis at that point: - Odd multiplicity means the graph crosses the axis. - Even multiplicity means the graph only touches the axis without crossing it.
• Real zeros are determined by solving the polynomial equation \( f(x) = 0 \).

For instance, given the problem's zeros \(-1, 2, \frac{10}{3}\), these are the x-values where the polynomial solution \( f(x) = 0 \).

These zeros are important for defining the polynomial as they significantly depict the function's interaction with the axes.

Leading Coefficient

The leading coefficient of a polynomial function is the coefficient of the term with the highest degree. In our context of third-degree polynomials, the term is the one with \( x^3 \).

This coefficient plays a key role in the behavior and shape of the polynomial graph:

• The absolute value of the leading coefficient impacts the vertical stretching or compression of the polynomial's graph.
• The sign of the leading coefficient determines the end behavior of the polynomial: - A positive leading coefficient means the graph extends to positive infinity as \( x \) goes to both negative and positive infinity. - A negative leading coefficient implies the graph heads toward negative infinity as \( x \) approaches negative and positive infinity.

For example, applying a positive or negative \( k \) value scaling, as in the step-by-step solution, results in two possible graphs, each displaying a distinct end behavior.

Understanding the influence of the leading coefficient is crucial when constructing and interpreting polynomial functions.