Problem 30 - 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. $$f(x)=-3\left(x+\frac{1}{2}\right)(x-4)^{3}$$

Step-by-Step Solution

Answer

The zeros of the function are \(x=-\frac{1}{2}\) (crosses x-axis) and \(x=4\) (touches x-axis and turns around). Their multiplicities are 1 and 3 respectively.

Step 1: Find Zeros

To find the zeros of \(f(x)=-3\left(x+\frac{1}{2}\right)(x-4)^{3}\), set \(f(x) = 0\) and solve for \(x\). This gives \(x+\frac{1}{2}=0\) and \(x-4=0\), from which we find that \(x=-\frac{1}{2}\) and \(x=4\).

Step 2: Determine Multiplicity of Each Zero

The exponent on each factor in the polynomial corresponds to the multiplicity of its corresponding zero. Thus, from the factors \(x+\frac{1}{2}\) and \((x-4)^{3}\), we can see that the zero -1/2 has multiplicity 1 and the zero 4 has multiplicity 3.

Step 3: Describe Behavior of the Graph at each Zero

At \(x=-\frac{1}{2}\), the graph will cross the x-axis, because the multiplicity of that zero is odd (1). At x=4, the graph will touch the x-axis and then turn around, because the multiplicity of that zero is also odd (3).

Key Concepts

Polynomial Zeros Multiplicity

Understanding the multiplicity of zeros in a polynomial function is crucial in analyzing the function's behavior. Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. This is determined by the exponent of the corresponding factor in the polynomial's factored form. For example, consider the polynomial function given by the equation \(f(x)=-3\left(x+\frac{1}{2}\right)(x-4)^{3}\). Here, the zero \(x=-\frac{1}{2}\) appears once, indicating it has a multiplicity of 1. Meanwhile, the zero \(x=4\) appears three times, as indicated by the exponent of 3 on the term \(x-4\), hence it has a multiplicity of 3.

Why is multiplicity important? It gives us insights into the graph of the polynomial function at each zero. For instance, if a zero's multiplicity is odd, the graph will cross the x-axis at that point. If the multiplicity is even, it will merely touch the x-axis and turn around, also known as 'bouncing off' the axis. Recognizing these patterns helps in sketching the rough graph of the polynomial and predicting its behavior without having to plot numerous points.

Graph Behavior at Zeros

The behavior of a polynomial graph at its zeros can be fascinating to study. As we've seen in the polynomial function \(f(x)=-3\left(x+\frac{1}{2}\right)(x-4)^{3}\), the graph behaves differently at \(x=-\frac{1}{2}\) compared to how it behaves at \(x=4\), due to the different multiplicities of these zeros. Since \(x=-\frac{1}{2}\) has a multiplicity of 1, which is odd, the graph will directly cross the x-axis at this point. Conversely, the zero at \(x=4\) has a multiplicity of 3, also odd, but higher than 1, which means the graph will still cross the x-axis but in a different manner - it will appear flatter near the x-axis and may exhibit a point of inflection where it 'turns around' right after crossing.

This behavior is key to visualizing how the graph of a polynomial shapes up between and beyond its zeros. Odd multiplicities lead to the graph crossing the x-axis, while even multiplicities lead to a 'bounce.' However, the steepness or gentleness of the graph's curve near these zeros will be influenced by the actual value of the multiplicity, with higher multiplicities generally resulting in more pronounced behavior.

Solving Polynomial Equations

Solving polynomial equations is a foundational skill in algebra that allows students to find the values for which the polynomial equals zero - these are known to be the roots or zeros of the equation. To solve a polynomial equation such as \(f(x)=-3\left(x+\frac{1}{2}\right)(x-4)^{3}=0\), the goal is to find the values of \((x)\) that make the equation true. This process involves setting the polynomial equal to zero and factoring it, then setting each factor equal to zero to solve for the zeros.

In the equation above, we factored the polynomial to find that the solutions are \(x=-\frac{1}{2}\) and \(x=4\). Solving polynomial equations becomes more challenging with higher-degree polynomials; however, the fundamental approach remains the same: factor where possible, apply the Zero Product Property, and solve each resulting equation. In some cases, where factoring is difficult or impossible, other methods such as polynomial division, synthetic division, or numerical methods may be employed to find an approximate or exact solution.