Questions: Find the zeros of the polynomial function and state the multiplicity of each. f(x)=-5(x-2)^4(x+5)^3 x^2 The smallest zero is with multiplicity .

Find the zeros of the polynomial function and state the multiplicity of each. f(x)=-5(x-2)^4(x+5)^3 x^2 The smallest zero is with multiplicity .
Transcript text: Find the zeros of the polynomial function and state the multiplicity of each. \[ f(x)=-5(x-2)^{4}(x+5)^{3} x^{2} \] The smallest zero is $\square$ with multiplicity $\square$ .
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Solution

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Solution Steps

Step 1: Identify the Factored Form

The polynomial is given in its factored form as: f(x) = -5(x - 2)^4(x + 5)^3(x - 0)^2.

Step 2: Find the Zeros

The zeros of the polynomial are: x = 2, x = -5, x = 0.

Step 3: Determine Multiplicities

The zero 2 has a multiplicity of 4, indicating the graph touches and turns around at this point. The zero -5 has a multiplicity of 3, indicating the graph crosses at this point. The zero 0 has a multiplicity of 2, indicating the graph touches and turns around at this point.

Final Answer:

The zero 2 has a multiplicity of 4. The zero -5 has a multiplicity of 3. The zero 0 has a multiplicity of 2.

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