Problem 26 - Algebra and Trigonometry

Question

In Exercises \(25-32,\) 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. $$f(x)=3(x+5)(x+2)^{2}$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x)=3(x+5)(x+2)^{2}\) are \(x=-5\) (with multiplicity 1) and \(x = -2\) (with multiplicity 2). The graph crosses the x-axis at \(x=-5\) and touches and turns around at \(x=-2\).

Step 1: Identify the Zeros of the Function

Given the function \( f(x)=3(x+5)(x+2)^{2}\). The zeros can be found by setting each factor equal to zero. Thus, the zeros are Yat \(x+5 = 0\) and \(x + 2 = 0\). These equations result in \(x = -5\) and \(x = -2\)

Step 2: Determine the Multiplicity of the Zeros

Multiplicity is given by the exponent of the factor. Looking at the function, the factor \(x+5\) has an exponent of 1, so it has a multiplicity of 1. The factor \(x+2\) has a an exponent of 2, so its multiplicity is 2. Thus we have a zero at \(x = -5\) with multiplicity 1 and a zero at \(x = -2\) with multiplicity 2.

Step 3: Analyze the Behavior at Each Zero

For a polynomial function, if a zero has an odd multiplicity, the graph will cross the x-axis at that zero. If a zero has an even multiplicity, the graph will touch the x-axis at that zero and turn around. Therefore, at \(x = -5\), the graph crosses the x-axis (odd multiplicity), and at \(x = -2\), the graph touches the x-axis and turns around (even multiplicity).

Key Concepts

Polynomial Function Analysis

Polynomial functions, such as
\( f(x)=3(x+5)(x+2)^{2} \)
play a pivotal role in mathematics, modeling a wide range of phenomena in sciences and engineering. To analyze these functions, one must understand their structure—they are algebraic expressions involving sums of powers in one or more variables multiplied by coefficients. A crucial aspect of polynomial function analysis is finding the function's zeros, which are the x-values where the function equals zero.

In the given exercise, the zeros are identified by factoring the polynomial and setting each factor equal to zero. The given solution guides step-by-step through this process, pointing out that the function has zeros when \( x = -5 \) and \( x = -2 \) by solving the equations \( x+5 = 0 \) and \( x+2 = 0 \) respectively. The concept of zeros is essential since it reveals where the polynomial intersects the x-axis, which has practical implications for problems including real-world applications, like physics and economics.

Multiplicity of Zeros

The multiplicity of a zero in a polynomial function refers to the number of times that the factor corresponding to the zero appears. It is an integer value indicated by the exponent of the factor when the polynomial is factored completely. Multiplicity tells us more than just the value of the zero; it gives insight into the function's behavior at that point on the graph.

As shown in the step-by-step solution, the zero \( x = -5 \) has a multiplicity of 1 because the factor \( x+5 \) appears once. In contrast, the zero \( x = -2 \) has a multiplicity of 2, as indicated by the squared factor \( (x+2)^{2} \). Understanding the concept of multiplicity is important not only for finding zeros but also for predicting the shape and turning points of the polynomial graph. Multiplicity influences how the graph behaves at each zero, affecting whether it crosses or just touches the x-axis.

Graph Behavior at Zeros

Knowing the zeros of a polynomial function and their associated multiplicities allows us to predict how the graph of the function will behave at those points. The graph of a polynomial function will cross the x-axis at zeros with an odd multiplicity and touch the x-axis at zeros with an even multiplicity before turning around.

As detailed in the exercise solution, the behavior at zero \( x = -5 \) is a crossing through the x-axis because its multiplicity is 1, an odd number. Conversely, at zero \( x = -2 \) the graph touches the x-axis and turns around due to its multiplicity of 2, an even number. This behavior is crucial for graph sketching and understanding the output values of the function in relation to its input values. It also aids in determining the number of real-world encounters or events at specific intervals, which can be particularly helpful in optimizing or predicting outcomes in various fields.