Problem 59 - College Algebra

Question

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-2 x^{2}-15 x$$

Step-by-Step Solution

Answer

The intercepts are (-3,0), (0,0), and (5,0), and the end behavior is as \( x \to \infty, \, f(x) \to \infty \) and as \( x \to -\infty, \, f(x) \to -\infty \).

Step 1: Analyze the Polynomial

The given polynomial is \( f(x) = x^3 - 2x^2 - 15x \). Notice that it is a cubic polynomial, which typically has up to 3 real roots and can have a maximum of two turning points due to its degree.

Step 2: Factor the Polynomial

To find the intercepts, first factor the polynomial. Start by factoring out the greatest common factor, which is \( x \):\[ f(x) = x(x^2 - 2x - 15) \]Continue factoring the quadratic \( x^2 - 2x - 15 \) using the AC method or trial and error: it factors to \((x-5)(x+3)\). So, the full factorization is:\[ f(x) = x(x - 5)(x + 3) \]

Step 3: Determine the Intercepts

The x-intercepts are the roots of the polynomial, given by setting \( f(x) = 0 \) and solving:\[ x(x - 5)(x + 3) = 0 \]This gives solutions \( x = 0 \), \( x = 5 \), and \( x = -3 \). The y-intercept is obtained by evaluating \( f(0) \), yielding \( f(0) = 0 \). Hence the function intercepts at (0,0), (5,0), and (-3,0).

Step 4: Use a Graphing Calculator

Enter \( f(x) = x^3 - 2x^2 - 15x \) into the graphing calculator. Use the graph to verify the intercepts: the function should cross the x-axis at \(-3\), \(0\), and \(5\).Ensure the window settings are adjusted to view all the intercepts and the overall shape of the curve.

Step 5: Determine the End Behavior

Since the polynomial is cubic, its end behavior can be described as follows: as \( x \rightarrow \, \infty \), \( f(x) \rightarrow \, \,\infty \), and as \( x \rightarrow \, - \infty \), \( f(x) \rightarrow \, - \infty \). This behavior is due to the highest degree term, \( x^3 \), which dominates the behavior of the function at extreme x values.

Key Concepts

Cubic Polynomials

Cubic polynomials are a fascinating type of polynomial with a degree of three. This means that the highest power of the variable, usually denoted as \( x \), is three. Unlike quadratic equations, which have a maximum of two solutions (or roots), cubic polynomials can have up to three real roots. This is because they can cross the x-axis up to three times. Due to their degree, cubic polynomials can have up to two turning points. A turning point is where the graph changes direction from increasing to decreasing or vice versa. These characteristics make cubic polynomials rich and complex, yet solvable, providing insights into their graphical representation and the nature of polynomial roots.

X-Intercepts

The x-intercepts of a polynomial function are the points at which the graph crosses the x-axis. These intercepts are essentially the solutions to the equation when the function \( f(x) = 0 \). For the cubic polynomial function \( f(x) = x^3 - 2x^2 - 15x \), the x-intercepts can be directly found using its factored form \( f(x) = x(x-5)(x+3) \). Setting \( f(x) = 0 \), the solutions are \( x = 0 \), \( x = 5 \), and \( x = -3 \).Each intercept corresponds to a root of the polynomial, and they indicate where the function changes sign—moving from positive to negative or negative to positive values.

End Behavior

End behavior analysis is crucial to understanding how a polynomial function behaves as the input \( x \) becomes very large (positive or negative). For cubic polynomials like \( f(x) = x^3 - 2x^2 - 15x \), the highest degree term \( x^3 \) guides this behavior as \( x \to \pm \infty \). Here’s how it works:

• As \( x \to \infty \), \( f(x) \) heads towards \( \,\infty \) because \( x^3 \) grows very large and positive.
• Conversely, as \( x \to -\infty \), \( f(x) \) tends toward \( -\infty \) since the cube of a negative number is negative and increasing in magnitude.

This simple pattern of growth and decay helps sketch the function’s graph, providing teachers and students with essential insights into the overall "shape" of the graph.

Factoring Polynomials

Factoring polynomials is a key method used to simplify polynomial equations and find the roots. It involves expressing the polynomial as a product of simpler polynomials that, when multiplied, recreate the original function. In the case of \( f(x) = x^3 - 2x^2 - 15x \), factoring begins by identifying the greatest common factor, which is \( x \). So, we factor it to \( x(x^2 - 2x - 15) \). Next, we focus on factoring the quadratic \( x^2 - 2x - 15 \). By trial and error, or using the AC method, this can be factored to \( (x-5)(x+3) \), resulting in the full factorization \( f(x) = x(x - 5)(x + 3) \).Factoring is valuable not only for finding zeros of polynomials but also for simplifying expressions for easier manipulation and clearer insight into their structure.