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Determine the zeros of the given polynomial function and the multiplicity of each zero. List the zeros from smallest to largest. If the zero is not an integer then use a fraction (not a decimal).F(x)=(3x+2)^5(x^2-10x+25)1st zero, x= Answer for part 1 with a multiplicity of Answer for part 22nd zero, x= Answer for part 3with a multiplicity of Answer for part 4

Determine the zeros of the given polynomial function and the multiplicity of each-example-1

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Answer:

• 1st zero: x=-2/3 with a multiplicity of 5.

,

• 2nd zero: x=5 with a multiplicity of 2.

Explanation:

Given the function:


F\mleft(x\mright)=\mleft(3x+2\mright)^5\mleft(x^2-10x+25\mright)

First, factorize the quadratic expression: x²-10x+25


\begin{gathered} x^2-10x+25=x^2-5x-5x+25 \\ =x(x-5)-5(x-5) \\ =(x-5)(x-5) \end{gathered}

Therefore, we can rewrite F(x) as:


F(x)=(3x+2)^5(x-5)^2

Solving for the zeroes:


\begin{gathered} (3x+2)^5=0 \\ \text{Subtract 2 from both sides} \\ 3x=-2 \\ \text{Divide both sides by 3} \\ x=-(2)/(3)\text{ (5 times)} \end{gathered}

• 1st zero: x=-2/3 with a multiplicity of 5.


\begin{gathered} (x-5)^2=0 \\ x=5\text{ (twice)} \end{gathered}

• 2nd zero: x=5 with a multiplicity of 2.

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