Questions: Find the zeros of the polynomial function, and state the multiplicity of each. f(x)=-3(x+5)(x+5)(x+5)(x-2)

Find the zeros of the polynomial function, and state the multiplicity of each. f(x)=-3(x+5)(x+5)(x+5)(x-2)
Transcript text: Find the zeros of the polynomial function, and state the multiplicity of each. \[ f(x)=-3(x+5)(x+5)(x+5)(x-2) \]
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Solution

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Solution Steps

Step 1: Identify the Polynomial Function

The given polynomial function is:

\[ f(x) = -3(x+5)(x+5)(x+5)(x-2) \]

This can be rewritten as:

\[ f(x) = -3(x+5)^3(x-2) \]

Step 2: Find the Zeros of the Polynomial

To find the zeros of the polynomial, set \( f(x) = 0 \):

\[ -3(x+5)^3(x-2) = 0 \]

This equation is satisfied when either \( (x+5)^3 = 0 \) or \( (x-2) = 0 \).

Solving \( (x+5)^3 = 0 \):

\[ x+5 = 0 \implies x = -5 \]

Solving \( (x-2) = 0 \):

\[ x-2 = 0 \implies x = 2 \]

Step 3: Determine the Multiplicity of Each Zero

The multiplicity of a zero is determined by the exponent of the factor in the polynomial.

  • The zero \( x = -5 \) comes from the factor \( (x+5)^3 \), so its multiplicity is 3.
  • The zero \( x = 2 \) comes from the factor \( (x-2) \), so its multiplicity is 1.

Final Answer

The zeros of the polynomial function are:

  • \( x = -5 \) with multiplicity 3
  • \( x = 2 \) with multiplicity 1

\[ \boxed{x = -5 \text{ (multiplicity 3)}, \, x = 2 \text{ (multiplicity 1)}} \]

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