Questions: Find the zeros for the polynomial function and give the multiplicity for the x -axis or touches the x -axis and turns around at each zero. f(x)=-2(x-5)(x+6)^3 Determine the zero(s). The zero(s) is/are

Find the zeros for the polynomial function and give the multiplicity for the x -axis or touches the x -axis and turns around at each zero. f(x)=-2(x-5)(x+6)^3 Determine the zero(s). The zero(s) is/are
Transcript text: Find the zeros for the polynomial function and give the multiplicity for the x -axis or touches the x -axis and turns around at each zero. \[ f(x)=-2(x-5)(x+6)^{3} \] Determine the zero(s). The zero(s) is/are $\square$
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Solution

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Solution Steps

To find the zeros of the polynomial function \( f(x) = -2(x-5)(x+6)^3 \), 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.

Solution Approach
  1. Set each factor of the polynomial equal to zero: \( x-5 = 0 \) and \( (x+6)^3 = 0 \).
  2. Solve these equations to find the zeros.
  3. Determine the multiplicity of each zero based on the exponent of the factor.
Step 1: Identify the Zeros of the Polynomial

The given polynomial function is:

\[ f(x) = -2(x-5)(x+6)^3 \]

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

\[ -2(x-5)(x+6)^3 = 0 \]

Step 2: Solve for Each Factor

The product is zero if any of the factors is zero. Therefore, we solve:

  1. \( x - 5 = 0 \)
  2. \( (x + 6)^3 = 0 \)
Solving \( x - 5 = 0 \):

\[ x = 5 \]

Solving \( (x + 6)^3 = 0 \):

\[ x + 6 = 0 \implies x = -6 \]

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 \( (x-5) \), which has an exponent of 1. Therefore, the multiplicity of \( x = 5 \) is 1.
  • The zero \( x = -6 \) comes from the factor \( (x+6)^3 \), which has an exponent of 3. Therefore, the multiplicity of \( x = -6 \) is 3.

Final Answer

The zeros of the polynomial are \( x = 5 \) and \( x = -6 \). The multiplicity of \( x = 5 \) is 1, and the multiplicity of \( x = -6 \) is 3.

\[ \boxed{x = 5, \, x = -6} \]

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