Questions: Polynomial and Rational Functions Finding zeros and their multiplicities given a polynomial function written i... Suppose that the polynomial function (f) is defined as follows. [ f(x)=9(x-6)^2(x-8)^3(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:

Polynomial and Rational Functions
Finding zeros and their multiplicities given a polynomial function written i...

Suppose that the polynomial function (f) is defined as follows.
[
f(x)=9(x-6)^2(x-8)^3(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:
Transcript text: Polynomial and Rational Functions Finding zeros and their multiplicities given a polynomial function written i... Suppose that the polynomial function $f$ is defined as follows. \[ f(x)=9(x-6)^{2}(x-8)^{3}(x+5)^{3} \] 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." \begin{tabular}{|ll|} \hline Zero(s) of multiplicity one: & $\square$ \\ Zero(s) of multiplicity two: & $\square$ \\ Zero(s) of multiplicity three: & $\square$ \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Factor the Polynomial

The polynomial function \( f(x) = 9(x-6)^{2}(x-8)^{3}(x+5)^{3} \) has been factorized into the form: \[ f(x) = 9 \left(x - 8\right)^{3} \left(x - 6\right)^{2} \left(x + 5\right)^{3} \]

Step 2: Identify the Zeros and Their Multiplicities

From the factorized form, we can identify the zeros of the polynomial and their corresponding multiplicities:

  • The zero at \( x = 6 \) has a multiplicity of \( 2 \).
  • The zero at \( x = 8 \) has a multiplicity of \( 3 \).
  • The zero at \( x = -5 \) has a multiplicity of \( 3 \).
Step 3: Categorize the Zeros

Now, we categorize the zeros based on their multiplicities:

  • Zero(s) of multiplicity one: \( \text{None} \)
  • Zero(s) of multiplicity two: \( 6 \)
  • Zero(s) of multiplicity three: \( 8, -5 \)

Final Answer

  • Zero(s) of multiplicity one: \(\boxed{\text{None}}\)
  • Zero(s) of multiplicity two: \(\boxed{6}\)
  • Zero(s) of multiplicity three: \(\boxed{8, -5}\)
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