We are given the following zeros of a polynomial
![0,-3,\sqrt[]{2}](https://img.qammunity.org/2023/formulas/mathematics/college/fy5y83h0e70hxnmp51lu3e57dlhfeuopw8.png)
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
![(x)(x+3)(x+\sqrt[]{2})(x-\sqrt[]{2})](https://img.qammunity.org/2023/formulas/mathematics/college/hbm8216c04auqpruotnzo0dl3hqiurfadh.png)
Now, let us expand and simplify the polynomial
![\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}](https://img.qammunity.org/2023/formulas/mathematics/college/bg3extfv8iuxgnib828vzof2j0se2hbd4r.png)
Therefore, the correct polynomial is the first option.
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