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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).G(x)=3x^4+6x^3+3x^21st zero = Answer for part 1 with a multiplicity of Answer for part 22nd zero = 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
User Icza
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If a polynomial can be factored as the product of k linear factors with exponents n₁, n₂, ..., n_k, as follows:


P(x)=C\cdot(x-a_1)^(n_1)*(x-a_2)^(n_2)*\ldots*(x-a_k)^(n_k)

Then, the zeros of P(x) are a₁, a₂, ...,a_k and their respective multiplicities are n₁, n₂, ..., n_k.

For the given polynomial, notice that 3x^2 is a common factor for all three terms. Then, factor it out:


\begin{gathered} G(x)=3x^4+6x^3+3x^2 \\ \Rightarrow G(x)=3x^2(x^2+2x+1) \end{gathered}

The factor (x^2+2x+1) is a perfect square binomial that can be expressed as a binomial squared:


x^2+2x+1=(x+1)^2

Then, G(x) can be written as:


\begin{gathered} G(x)=3x^2(x+1)^2 \\ \Rightarrow G(x)=3(x-0)^2(x-\lbrack-1\rbrack)^2 \end{gathered}

As we can see from the expression, the roots of G(x) from the smallest to largest are -1 and 0, and the multiplicities are 2 in both cases.

Therefore, the first zero is -1 with a multiplicity of 2, and the second zero is 0 with a multiplicity of 2.

User AntoineMoPa
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