Questions: Suppose that the polynomial function (f) is defined as follows. [ f(x)=(x-8)(x-11)(x+12)^2(x+5)^3 ] List each zero of (f) according to its multiplicity in the categories below. If there is more than one answer for a multiplicity, separate them with commas. If there is no answer, click on "None." Zero(s) of multiplicity one: Zero(s) of multiplicity two: Zero(s) of multiplicity three:

Solution Steps

To find the zeros of the polynomial function and their multiplicities, we need to examine each factor of the polynomial. The zeros are the values of \(x\) that make each factor equal to zero. The multiplicity of each zero is determined by the exponent of the corresponding factor.

1. Identify the factors of the polynomial: \((x-8)\), \((x-11)\), \((x+12)^2\), and \((x+5)^3\).
2. Determine the zero for each factor:
   • \((x-8)\) gives a zero at \(x=8\) with multiplicity 1.
   • \((x-11)\) gives a zero at \(x=11\) with multiplicity 1.
   • \((x+12)^2\) gives a zero at \(x=-12\) with multiplicity 2.
   • \((x+5)^3\) gives a zero at \(x=-5\) with multiplicity 3.
3. List the zeros according to their multiplicities.

The given polynomial function is:

\[ f(x) = (x-8)(x-11)(x+12)^{2}(x+5)^{3} \]

To find the zeros of the polynomial, we set \( f(x) = 0 \). The zeros are the values of \( x \) that make each factor equal to zero.

1. Zero from \((x-8)\):

   • Set \( x-8 = 0 \) to find the zero.
   • Solving gives \( x = 8 \).
   • The multiplicity of this zero is 1 because the factor \((x-8)\) appears once.

2. Zero from \((x-11)\):

   • Set \( x-11 = 0 \) to find the zero.
   • Solving gives \( x = 11 \).
   • The multiplicity of this zero is 1 because the factor \((x-11)\) appears once.

3. Zero from \((x+12)^{2}\):

   • Set \( x+12 = 0 \) to find the zero.
   • Solving gives \( x = -12 \).
   • The multiplicity of this zero is 2 because the factor \((x+12)\) is squared.

4. Zero from \((x+5)^{3}\):

   • Set \( x+5 = 0 \) to find the zero.
   • Solving gives \( x = -5 \).
   • The multiplicity of this zero is 3 because the factor \((x+5)\) is cubed.

Final Answer