Math Problem Statement

What could be the equation of the polynomial p(x) based on the graph provided?

Solution

To identify the correct equation of the polynomial p(x)p(x) from the graph:

  1. Identify the x-intercepts: The graph intersects the x-axis at x=1x = -1, x=2.5x = -2.5, and x=3x = 3. These are the roots of the polynomial.

  2. Check the behavior at each intercept:

    • At x=1x = -1, the graph touches the x-axis and turns around, indicating a double root.
    • At x=2.5x = -2.5, the graph crosses the x-axis, indicating a single root.
    • At x=3x = 3, the graph touches the x-axis and turns around, indicating a double root.
  3. Determine the corresponding factors:

    • (x+1)2(x + 1)^2 for the double root at x=1x = -1.
    • (2x+5)(2x + 5) for the single root at x=2.5x = -2.5.
    • (x3)2(x - 3)^2 for the double root at x=3x = 3.
  4. Choose the correct equation: The polynomial equation that fits these roots and their multiplicities is: p(x)=(x+1)2(2x+5)(x3)2p(x) = (x + 1)^2 (2x + 5)(x - 3)^2

This corresponds to option A in the provided choices.

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Related Questions:

  1. How do you determine if a root is a double root or a single root from a graph?
  2. What is the significance of the multiplicity of roots in polynomial graphs?
  3. How does the leading coefficient affect the end behavior of the polynomial graph?
  4. What happens when a polynomial graph has a triple root at a point?
  5. Can a polynomial have complex roots, and how would they affect the graph?

Tip:

Always verify the behavior of the graph at each intercept to confirm the multiplicity of the roots when determining polynomial equations.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Roots of Polynomials
Multiplicities of Roots

Formulas

Polynomial factorization based on roots
Double root and single root behavior

Theorems

Factor Theorem
Multiplicity of Roots Theorem

Suitable Grade Level

Grades 10-12