Question 5
Find all zeros of $f(x)=x^{3}+x^{2}-8 x-6$. Enter the zeros separat decimal approximations.
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Answer
The complex zeros of the polynomial function \(f(x)=x^{3}+x^{2}-8 x-6\) are \(\boxed{-3}\), \(\boxed{1 - \sqrt{2}}\), and \(\boxed{1 + \sqrt{2}}\).
Step 1 ：Set the function \(f(x)=x^{3}+x^{2}-8 x-6\) equal to zero and solve for x to find the real zeros of the polynomial function.
Step 2 ：The real zeros of the polynomial are \(-3\), \(1 - \sqrt{2}\), and \(1 + \sqrt{2}\).
Step 3 ：Since these zeros are already complex, there is no need to perform synthetic division and solve a quadratic equation.
Step 4 ：Therefore, the complex zeros of the polynomial are \(-3\), \(1 - \sqrt{2}\), and \(1 + \sqrt{2}\).
Step 5 ：The complex zeros of the polynomial function \(f(x)=x^{3}+x^{2}-8 x-6\) are \(\boxed{-3}\), \(\boxed{1 - \sqrt{2}}\), and \(\boxed{1 + \sqrt{2}}\).
