Questions: Find the zeros of the polynomial function and state the multiplicity of each. f(x)=-5 x^2(x-7)(x+4)^3

Solution Steps

To find the zeros of the polynomial function, we need to identify the values of \( x \) that make the function equal to zero. This involves setting each factor of the polynomial to zero and solving for \( x \). The multiplicity of each zero is determined by the exponent of the corresponding factor.

The given polynomial is:

\[ f(x) = -5x^2(x-7)(x+4)^3 \]

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

\[ -5x^2(x-7)(x+4)^3 = 0 \]

The zeros occur when each factor is equal to zero:

1. \( x^2 = 0 \)
2. \( x - 7 = 0 \)
3. \( (x + 4)^3 = 0 \)

1. Solve \( x^2 = 0 \):

   \[ x = 0 \]

2. Solve \( x - 7 = 0 \):

   \[ x = 7 \]

3. Solve \( (x + 4)^3 = 0 \):

   \[ x + 4 = 0 \implies x = -4 \]

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

1. Multiplicity of \( x = 0 \):

   The factor \( x^2 \) indicates a multiplicity of 2.

2. Multiplicity of \( x = 7 \):

   The factor \( x - 7 \) indicates a multiplicity of 1.

3. Multiplicity of \( x = -4 \):

   The factor \( (x + 4)^3 \) indicates a multiplicity of 3.

Final Answer