Problem 28 - 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. $$f(x)=3(x+5)(x+2)^{2}$$

Step-by-Step Solution

Answer

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

Step 1 - Find the zeros

To find the zeros or roots of the function \(f(x) = 3(x+5)(x+2)^{2}\), we set the function equal to zero: \(3(x+5)(x+2)^{2} = 0\). To solve for \(x\), each factor set to zero. So \(x+5 = 0\) or \((x+2)^{2} = 0\). This yields \(x=-5\) and \(x=-2\).

Step 2 - Determine the multiplicity

The multiplicity of a root is the number of times it appears as a factor. The factor \(x+5\) appears once, so the root \(x=-5\) has a multiplicity of 1. The factor \((x+2)^{2}\) appears with exponent 2 and the root \(x=-2\) has a multiplicity of 2.

Step 3 - Determine whether the graph crosses the x-axis

For the root \(x=-5\), the multiplicity is odd (1), so the graph of the function crosses the \(x\)-axis at \(x=-5\). For the root \(x=-2\), the multiplicity is even (2), so the graph touches the \(x\)-axis at \(x=-2\) and turns around.

Key Concepts

Polynomial Roots

When studying polynomial functions, one of the fundamental tasks is finding the points where the graph intersects the horizontal axis. These points are known as the polynomial roots or zeros. To locate these roots, we look for values of the variable, usually x, that make the function equal to zero. In algebra, this process often involves factoring the polynomial and setting each factor equal to zero, because of the Zero Product Property, which states that if a product equals zero, at least one of the factors must be zero.

For the polynomial function given by f(x) = 3(x+5)(x+2)^2, the roots can be found by solving for the values that make each factor (x+5) and (x+2) equal to zero, resulting in x = -5 and x = -2. Understanding where and how to locate these zeros is crucial for graphing the polynomial and for further applications in calculus and beyond.

Root Multiplicity

The multiplicity of a root in a polynomial function reflects how many times that particular root occurs as a factor. It provides insight into the function's behavior around the root on a graph and has implications for both algebraic and calculus concepts. Multiplicity can be determined by examining the exponent on the factor associated with the root in the factored form of the function.

To illustrate this concept, consider the same function f(x) = 3(x+5)(x+2)^2. The root x = -5 corresponds to the factor (x+5) and has a multiplicity of 1 since it appears only once. Conversely, the root x = -2 corresponds to the factor (x+2) squared and thus has a multiplicity of 2. A root's multiplicity affects not only the touch or cross behavior at the x-axis but also plays a role in determining the shape and the curvature of the graph near the root.

Graph Behavior at Roots

The graph behavior of a polynomial at its roots is directly tied to the multiplicity of those roots. A polynomial's graph will either cross or touch the x-axis at each root, and this behavior can be predicted by the root's multiplicity. An odd multiplicity indicates that the graph of the polynomial will cross the x-axis, creating an intersection at that point. In contrast, an even multiplicity suggests that the graph of the polynomial touches the x-axis and turns around, without crossing it.

Based on the previously analyzed function f(x), at x = -5, with a multiplicity of 1, an odd number, the graph will cross the x-axis. Meanwhile, at x = -2, with a multiplicity of 2, an even number, the graph will touch the x-axis and revert direction. Recognizing these patterns is beneficial for sketching accurate graphs and for understanding the nuanced behavior of polynomials.

Algebraic Problem-Solving

Algebraic problem-solving involves a systematic approach to working with equations and formulas to find solutions. When dealing with polynomials, the same structured methods apply. The first step is typically to express the polynomial in its simplest form, often by factoring. Next, problem-solvers apply mathematical properties and theorems, like the Zero Product Property, to find variable values that satisfy the equation. This process requires careful manipulation of algebraic expressions and recognition of patterns and relationships.

In the context of the given example, the solution process began with setting the function equal to zero, followed by factoring and then applying the Zero Product Property to find the zeros. Each subsequent step involved understanding the implications of multiplicity and graph behavior. Strengthening these algebraic problem-solving skills is paramount for students as they build the foundation necessary for advanced mathematical applications.