Problem 34 - College Algebra

Question

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+5 x^{2}-9 x-45$$

Step-by-Step Solution

Answer

The zeros of the polynomial function \( x^{3} + 5x^{2} - 9x - 45 \) are \( x = 3, -3, -5 \), all with a multiplicity of 1. The graph of the function crosses the x-axis at each zero.

Step 1: Set the function equal to zero

This gives the equation \( x^{3} + 5x^{2} - 9x - 45 = 0 \).

Step 2: Solve the equation for x

By factoring the polynomial, we obtain \( (x - 3)(x + 3)(x + 5) = 0 \). The zeros of the polynomial are then the solutions of this equation, which can be found by setting each factor equal to zero. This gives \( x = 3, -3, -5 \).

Step 3: Find the multiplicity of each zero

As each zero appears only once in the factorization, the multiplicity of each zero is 1.

Step 4: Determine if the graph crosses or touches the x-axis at each zero

As the multiplicity of each zero is 1 (an odd integer), the graph of the function crosses the x-axis at each zero.

Key Concepts

Finding Zeros of Polynomials

When working with polynomial functions, finding the zeros is a fundamental task. A zero of a polynomial is any value of the independent variable, often denoted as `x`, that will make the polynomial equal to zero. Think of these points as places where the graph of the polynomial touches or crosses the x-axis.

In our example with the polynomial function \( f(x) = x^3 + 5x^2 - 9x - 45 \), we set the equation equal to zero to find the points at which the graph will intercept the x-axis. Factoring the polynomial, we got the factors \( (x - 3)(x + 3)(x + 5) \). Thus, setting each factor to zero, we find the zeros to be \( x = 3, -3, \) and \( -5 \).

Strategies for Finding Zeros:

• Factoring the polynomial directly, if possible
• Using the Rational Root Theorem to find potential rational zeros
• Applying synthetic division or long division
• Employing numerical methods such as the Newton-Raphson method when zeros cannot be easily found

Various strategies can be used depending on the complexity of the polynomial. Understanding these methods can help you navigate through tougher problems with ease.

Polynomial Root Multiplicity

Root multiplicity refers to the number of times a particular root appears as a factor of the polynomial when it is factored completely. The multiplicities give us important information about the graph of the polynomial, especially concerning its interaction with the x-axis when plotted.

In the given problem, \( f(x) = x^3 + 5x^2 - 9x - 45 \), each zero appears once, indicating that all zeros have a multiplicity of 1. This indicates that the graph will cross the x-axis at those points rather than just touching it and turning around. If a zero had, for instance, a multiplicity of 2, we would expect the graph to touch the x-axis at that zero and turn around.

Implications of Multiplicity:

• A zero with an odd multiplicity will mean the graph crosses the x-axis at that point.
• A zero with an even multiplicity means the graph touches the x-axis and turns around at that point.

The concept of multiplicity is not just a mathematical curiosity but has practical implications in graph analysis and calculus.

Graph Behavior at Zeros

The behavior of a polynomial graph near its zeros is crucial for graph sketching and understanding the function's overall shape. As seen in the exercise with \( f(x) = x^3 + 5x^2 - 9x - 45 \), knowing that all the zeros have a multiplicity of one tells us that the graph will cross the x-axis at the points \( x = 3, -3, \) and \( -5 \).

Further graphical behavior can be deduced from the leading coefficients and the highest power of the polynomial, which in this case suggests that the graph will extend upwards on both ends since the leading coefficient (1, from the term \( x^3 \)) is positive, and the degree is odd.

Key Aspects of Graph Behavior at Zeros:

• Crossing behavior occurs at zeros with odd multiplicity.
• Touching and turning behavior occurs at zeros with even multiplicity.
• The degree and leading coefficient of the polynomial provide information about the end behavior of the graph.

By analyzing these factors, students can gain a more intuitive understanding of how polynomials behave and how to predict their graphs accurately.