Verified Answer :

  • Identify the factors of the polynomial from the given expression: \((x-3)^2\), \((x+2)^2\), and \((x-1)\).
  • Determine the zero for each factor and associate it with the exponent (the multiplicity).
  • Identify that the factor \((x-1)\) has an exponent of 1, making \(x=1\) the zero with multiplicity 1.
  • Identify that the factor \((x+2)^2\) has an exponent of 2, making the multiplicity of the zero \(-2\) equal to 2.
  • The zero \(\boxed{1}\) has a multiplicity of \(1\).
  • The zero \(-2\) has a multiplicity of \(\boxed{2}\).

Explanation

  1. Analyzing the Factors and Zeros

The function is given by \(f(x) = (x-3)^2(x+2)^2(x-1)\). To find the zeros and their multiplicities, we examine the factored form of the polynomial. A zero \(c\) corresponds to a factor \((x-c)^n\) in the polynomial, where \(n\) represents the multiplicity of that zero. Our expression is already in factored form:
\[f(x) = (x-3)^2 \cdot (x+2)^2 \cdot (x-1)^1\]
We can identify the roots by setting each factor equal to zero:

  1. \(x-3 = 0 \Rightarrow x = 3\), with multiplicity \(2\).
  2. \(x+2 = 0 \Rightarrow x = -2\), with multiplicity \(2\).
  3. \(x-1 = 0 \Rightarrow x = 1\), with multiplicity \(1\).
  1. Identifying the Required Values

From our analysis of the factors, we identified the zeros and their respective multiplicities:

  • The zero \(x=1\) is associated with the factor \((x-1)^1\), so its multiplicity is \(1\).
  • The zero \(x=-2\) is associated with the factor \((x+2)^2\), so its multiplicity is \(2\).

Therefore, the zero with multiplicity \(1\) is \(1\), and the multiplicity of the zero \(-2\) is \(2\).

  1. Conclusion

We have determined that the zero with multiplicity \(1\) is \(1\), and the multiplicity of the zero \(-2\) is \(2\).

The zero \(\boxed{1}\) has a multiplicity of \(1\).

The zero \(-2\) has a multiplicity of \(\boxed{2}\).

Examples

In engineering and physics, the concept of multiplicity is essential when analyzing the stability of a system. For instance, if you are modeling the resonance of a mechanical structure, the zeros of a characteristic polynomial represent natural frequencies. A root with multiplicity 1 suggests a simple crossing point, whereas a root with multiplicity 2 or higher (like \(-2\) in our problem) indicates a "tangent" contact point, which can be critical for understanding how the system behaves under external vibrations or loads. By identifying these multiplicities, engineers can predict if a system will remain stable or potentially experience failure under specific frequencies.

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