Problem 101 - Algebra and Trigonometry

Question

In Exercises 100–103, determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed \(f(x)=(x+2)^{3}(x-4)^{2},\) and the graph touched the \(x\) -axis and turned around at \(-2\)

Step-by-Step Solution

Answer

The statement does not make sense because, for the function \(f(x)=(x+2)^{3}(x-4)^{2}\), the graph should cross the x-axis at x=-2, not just touch it and turn around, due to the odd multiplicity of the root -2.

Step 1: Identify Roots

Identify the roots of a polynomial, which are obtained by suitably factoring the function \(f(x)=(x+2)^{3}(x-4)^{2}\). The roots -2 and 4 are found from the factors \((x+2)\) and \((x-4)\) respectively.

Step 2: Determine Multiplicity behavior

Look at the multiplicity of each root. The root -2 comes from the factor \((x+2)^3\), the exponent '3' is an odd number. A root with an odd multiplicity will cause the graph to cross the x-axis at the given root location. On the other hand, the root 4 comes from the factor \((x-4)^2\), the exponent '2' is an even number. A root with even multiplicity will cause the graph to touch the x-axis at that point, and then reverse its direction.

Step 3: Verify the Statement

Verify the provided statement: 'I graphed \(f(x)=(x+2)^{3}(x-4)^{2}\), and the graph touched the x-axis and turned around at -2.' This statement is incorrect. Based on the information in Step 2, it is clear that the graph should cross the x-axis at x=-2 (due to the odd multiplicity of 3), not just touch it.

Key Concepts

Roots of Polynomial

The roots of a polynomial function are the values of the variable that make the entire function equal to zero. Essentially, these are the points where the graph of the polynomial intersects or touches the x-axis. In the given function, \(f(x) = (x+2)^{3}(x-4)^{2}\), we can find the roots by setting each factor of the function equal to zero and solving:

• \(x + 2 = 0 \Rightarrow x = -2\)
• \(x - 4 = 0 \Rightarrow x = 4\)

Thus, the roots of the polynomial are \(-2\) and \(4\). Each root arises from a factor in the polynomial, with the exponent of the factor hinting at how the graph will behave at these specific points. Recognizing these roots is a fundamental step in understanding and plotting polynomial graphs.

Multiplicity of Roots

The multiplicity of a root refers to the number of times a given root is repeated in a polynomial equation. It is determined by the power of the factor associated with that root in the polynomial's factored form. This affects the graph's interaction with the x-axis:

• If a root is associated with an odd power, the graph will intersect or cut through the x-axis at that point. For example, the root \(-2\) has a multiplicity of 3 (since the factor is \((x+2)^3\)), indicating that the graph will cross the x-axis at \(x = -2\).


• Conversely, if a root is associated with an even power, the graph will touch or bounce off the x-axis and not cross it. This is seen with the root \(4\), which has a multiplicity of 2 (from the factor \((x-4)^2\)), meaning the graph will just touch the x-axis at \(x = 4\) and reverse direction.

Understanding the multiplicity of roots is crucial for anticipating the shape and behavior of polynomial graphs across the x-axis.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting the curve of the function to see where it intersects with or touches the x-axis, and understanding its general shape. For a proper graph of the polynomial \(f(x) = (x+2)^{3}(x-4)^{2}\):

• Recognize the roots \(-2\) and \(4\). These are your key points of interaction with the x-axis.
• Check the multiplicity of each root:
  • At \(x = -2\), expect the graph to cross due to odd multiplicity (3).
  • At \(x = 4\), the graph will touch and turn back due to even multiplicity (2).
• Consider the end behavior. Since the highest degree of the polynomial is 5 (3 from \(x+2\) and 2 from \(x-4\)), the ends of the graph will move in opposite directions: one side will go to positive infinity and the other to negative infinity as \(x\) increases or decreases.

Graphing enables us to visualize the complex behavior of polynomial equations practically, showing how roots and their multiplicity directly influence the curve's path.