Find all zeros of f

f(t)=(t-2)^3-(t-2)

ok... now do I have to factor out (t-2)^3 to [t^3-6(t^2)-12t-8] and then minus (t-2) resulting in [t^3-6(t^2)+13t-6]?
if so, where can I go from there??

sorry... the second answer would be [t^3-6(t^2)+11t-6]

To find the zeros of the function f(t) = (t-2)^3 - (t-2), you can start by factoring it as you mentioned:

f(t) = (t-2)^3 - (t-2)

However, there seems to be a small error in your second expression. The correct factorization would be:

f(t) = t^3 - 6t^2 + 11t - 6

Now, to find the zeros of the function, you need to set f(t) equal to zero and solve for t:

t^3 - 6t^2 + 11t - 6 = 0

One possible approach to find the zeros is by using algebraic techniques, such as factoring, synthetic division, or using the Rational Root Theorem. However, in this case, those methods might not be straightforward.

An alternative approach is to use numerical methods, such as graphing the function or using iterative algorithms like Newton's method or the bisection method to approximate the zeros. Let's use a graphing calculator as an example:

1. Graph the function f(t) = t^3 - 6t^2 + 11t - 6 on a graphing calculator or a graphing software.

2. Look for the x-intercepts (where the graph of the function crosses the x-axis). These x-intercepts represent the zeros of the function.

3. From the graph, you can visually estimate the values of these zeros. In this case, you should find three zeros, let's call them t1, t2, and t3.

4. Record the estimated values of the zeros.

Remember that these values are approximate, and we can verify them by substituting them back into the original equation f(t) = 0 to see if they satisfy the equation.

Using a graphing calculator or software can provide a quick visualization and estimation of the zeros. However, for a more precise result, using numerical methods like Newton's method or the bisection method would be recommended. These methods would involve iterative calculations to refine the approximations until a desired level of accuracy is achieved.