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question 7
select all x - coordinates of x - intercepts of the graph of the function (f(x)=x^{3}+x^{2}-8x - 8).
□ (sqrt{27})
□ - 2
□ - 3
□ (-3sqrt{3})
□ nothing in this list is correct.

Explanation:

Step1: Recall x - intercept definition

The x - intercepts of a function \(y = f(x)\) occur when \(y = 0\), so we set \(f(x)=x^{3}+x^{2}-8x - 8 = 0\).

Step2: Group the terms

Group the terms as \((x^{3}+x^{2})-(8x + 8)=0\). Then factor out the common factors from each group: \(x^{2}(x + 1)-8(x + 1)=0\).

Step3: Factor out \((x + 1)\)

We get \((x + 1)(x^{2}-8)=0\).

Step4: Solve for \(x\)

Using the zero - product property, if \(ab = 0\), then \(a=0\) or \(b = 0\). So \(x+1 = 0\) gives \(x=-1\), and \(x^{2}-8=0\) gives \(x=\pm\sqrt{8}=\pm2\sqrt{2}\). Since none of the given options \(\sqrt{27},-2,-3,-3\sqrt{3}\) match the solutions of the equation \(x^{3}+x^{2}-8x - 8 = 0\).

Answer:

Nothing in this list is correct.