We are given the following zeros of a polynomial
[tex]0,-3,\sqrt[]{2}[/tex]
Let us find the polynomial corresponding to these zeros.
Notice that one of the zeros (√2) is irrational which always comes in conjugate pairs
Let us write these zeros in the factored form
[tex](x)(x+3)(x+\sqrt[]{2})(x-\sqrt[]{2})[/tex]
Now, let us expand and simplify the polynomial
[tex]\begin{gathered} (x)(x+3)(x+\sqrt[]{2})(x-\sqrt[]{2}) \\ (x^2+3x)(x+\sqrt[]{2})(x-\sqrt[]{2}) \\ (x^2+3x)(x^2-\sqrt[]{2}x+\sqrt[]{2}x-\sqrt[]{2}\cdot\sqrt[]{2}) \\ (x^2+3x)(x^2-2) \\ x^4-2x^2+3x^3-6x \\ x^4+3x^3-2x^2-6x \end{gathered}[/tex]
Therefore, the correct polynomial is the first option.
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