Math Problem Statement

Use the graph to write the formula for a polynomial function of least degree.

Solution

The graph provided shows a polynomial function, and the task is to write its formula for the polynomial function of least degree.

Here’s how we can approach it:

Step 1: Identify the Zeros (Roots)

From the graph:

  • The function crosses the x-axis at approximately x=2x = -2, x=0x = 0, and x=3x = 3. Therefore, the zeros of the function are x=2x = -2, x=0x = 0, and x=3x = 3.

Step 2: Determine the Multiplicity of Each Root

  • At x=2x = -2, the graph passes through the axis, indicating it has a root of multiplicity 1.
  • At x=0x = 0, the graph also passes through the axis, which suggests this root also has a multiplicity of 1.
  • At x=3x = 3, the graph touches the axis and bounces back up, which indicates a root with an even multiplicity. A multiplicity of 2 is the simplest assumption for the least degree.

Step 3: Construct the Polynomial

Given the roots, the polynomial should include factors that correspond to each root:

  • For x=2x = -2, the factor is (x+2)(x + 2).
  • For x=0x = 0, the factor is xx.
  • For x=3x = 3, the factor is (x3)2(x - 3)^2.

Thus, the polynomial is of the form: f(x)=ax(x+2)(x3)2f(x) = a \cdot x \cdot (x + 2) \cdot (x - 3)^2 where aa is a constant to be determined.

Step 4: Estimate the Leading Coefficient

By looking at the behavior of the graph as xx \to \infty, the graph opens upwards, meaning aa is positive. To further refine aa, we can use the y-intercept (which appears to be at around f(0)=0f(0) = 0) to determine the precise value of aa, but for now, assume a=1a = 1 for simplicity.

Thus, the polynomial can be written as: f(x)=x(x+2)(x3)2f(x) = x(x + 2)(x - 3)^2

Would you like further details or explanations about any step?

Here are some additional questions to deepen understanding:

  1. How does the multiplicity of a root affect the shape of the graph at that root?
  2. What happens to the degree of a polynomial when a root has higher multiplicity?
  3. How can you verify if the leading coefficient is correct using another point on the graph?
  4. What are some methods for identifying the degree of a polynomial based on its graph?
  5. How would the graph behave differently if aa were negative?

Tip: Always double-check for symmetry in the graph when determining the form of the polynomial—this can give clues about the degree and behavior of the function.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Roots and Zeros
Multiplicity

Formulas

f(x) = a * (x + 2)(x)(x - 3)^2

Theorems

Root Multiplicity Theorem

Suitable Grade Level

Grades 11-12 (Precalculus/Calculus)