Math Problem Statement

The graph of y = P(x) is shown. Which of the following could define P(x)?

Solution

The image shows a problem asking which of the given polynomials could define the function P(x)P(x) represented by the graph.

Key observations from the graph:

  1. Roots (x-intercepts): The graph crosses the x-axis at x=3x = -3, x=0x = 0, and x=2x = 2.
  2. Behavior near the roots:
    • At x=0x = 0, the graph touches the x-axis and turns back up, indicating a repeated root or a double root.
    • At x=3x = -3 and x=2x = 2, the graph crosses the x-axis, indicating simple roots at these points.

Forming the polynomial based on this behavior:

  • Since the graph touches the x-axis at x=0x = 0, we have a double root at x=0x = 0, meaning a factor of x2x^2.
  • The simple roots at x=3x = -3 and x=2x = 2 correspond to factors (x+3)(x + 3) and (x2)(x - 2), respectively.

Thus, the polynomial should take the form: P(x)=(x+3)(x2)(x2)12P(x) = \frac{(x + 3)(x^2)(x - 2)}{12}

Checking the options:

  • Option B matches this form: P(x)=(x+3)(x2)(x2)12P(x) = \frac{(x + 3)(x^2)(x - 2)}{12}

Thus, the correct answer is B.

Would you like a deeper explanation of any step?

Here are five related questions:

  1. How do you identify simple and repeated roots from a graph?
  2. What is the effect of a double root on the shape of the graph?
  3. How would the polynomial change if the root at x=2x = 2 were a double root?
  4. How does factoring a polynomial help in graphing it?
  5. How does the leading coefficient affect the end behavior of a polynomial?

Tip: When analyzing graphs, always look at how the curve behaves near the x-intercepts to determine the multiplicity of the roots.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Graphing
Roots of Polynomials

Formulas

Factorized form of polynomial: P(x) = (x - r1)(x - r2)...(x - rn)

Theorems

Multiplicity of Roots Theorem

Suitable Grade Level

Grades 10-12