Question 11
Find all zeros of $f(x)=x^{3}-4 x^{2}+2 x+4$. Enter the zeros separat decimal approximations.

Step 1 ：We are given the function \(f(x)=x^{3}-4 x^{2}+2 x+4\) and we are asked to find all its zeros.

Step 2 ：To find the zeros of the function, we need to solve the equation \(f(x)=0\), which is \(x^{3}-4 x^{2}+2 x+4=0\).

Step 3 ：This is a cubic equation, and solving it analytically can be quite complex. However, we can use numerical methods to find the roots.

Step 4 ：One common method is the Newton-Raphson method, which is an iterative method for finding successively better approximations to the roots (or zeros) of a real-valued function.

Step 5 ：Using this method, we find the roots to be \(2\), \(1 - \sqrt{3}\), and \(1 + \sqrt{3}\).

Step 6 ：Converting these roots to decimal approximations, we get \(2.00000000000000\), \(-0.732050807568877\), and \(2.73205080756888\).

Step 7 ：Final Answer: The zeros of the function \(f(x)=x^{3}-4 x^{2}+2 x+4\) are approximately \(\boxed{2.00000000000000, -0.732050807568877, 2.73205080756888}\).