Math Problem Statement

Given the graph of a degree 5 polynomial, complete the table of values for either the x-value of a zero, or the multiplicity of the zero.

Solution

The graph shows a degree 5 polynomial, and you're asked to complete the table of values based on the x-values of the zeros and their corresponding multiplicities.

Analyzing the graph:

  1. The polynomial crosses the x-axis at x=4x = -4, indicating a zero at x=4x = -4.

    • Since the graph crosses the x-axis at this point, the multiplicity of this root is odd. We already see in the table that its multiplicity is missing.
  2. The polynomial also crosses the x-axis at x=1x = 1, and since it crosses straight through, the multiplicity is 1 (this value is already given in the table).

  3. The graph touches the x-axis at approximately x=1x = -1, meaning it doesn't cross but touches the x-axis and turns around, indicating a multiplicity greater than 1 (typically 2).

Table Completion:

  • The zero at x=4x = -4 has multiplicity 1 because it crosses the x-axis.
  • The root at x=1x = -1 should be added to the table with multiplicity 2.

Here's the completed table:

Root with x=x =Multiplicity
4-41
1-12
111

Would you like a more detailed explanation of how multiplicities work, or have any questions?


Here are 5 related questions:

  1. How can you tell if a root has an odd or even multiplicity just by looking at the graph?
  2. What would happen if a root has a multiplicity of 3?
  3. Can a polynomial of degree 5 have more than 5 roots?
  4. How does the degree of the polynomial affect the shape of the graph near a root with multiplicity greater than 1?
  5. How can you factor this polynomial based on the roots identified?

Tip: The degree of a polynomial dictates the maximum number of roots (real or complex) it can have.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Zeros of a Polynomial
Multiplicity of Roots

Formulas

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Theorems

Multiplicity Theorem for Polynomials

Suitable Grade Level

Grades 10-12