To find the zeros of the function $f(x) = x^3 - 5x^2 + 6x$, we need to solve for $x$ when $f(x) = 0$. That is, we need to solve the equation:

$x^3 - 5x^2 + 6x = 0$

Step 1: Factor the equation

First, notice that there is a common factor of $x$ in all three terms:

$x(x^2 - 5x + 6) = 0$

Step 2: Solve the factored equation

Now, we have two factors: $x = 0$ and $x^2 - 5x + 6 = 0$.

Solve $x = 0$:

This gives one root: $x = 0$.

Solve $x^2 - 5x + 6 = 0$:

We can factor this quadratic expression. We need two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. So, we can factor as:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

Step 3: Solve for $x$

Set each factor equal to zero:

1. $x - 2 = 0 \Rightarrow x = 2$
2. $x - 3 = 0 \Rightarrow x = 3$

Step 4: Conclusion

The solutions to the equation $f(x) = x^3 - 5x^2 + 6x = 0$ are:

$x = 0, \, x = 2, \, x = 3$

These are the zeros of the function.

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Would you like a more detailed explanation of the factoring steps, or do you have any other questions about this problem?

Related Questions:

1. How can we verify that $x = 0, 2, 3$ are the correct solutions?
2. What are the general methods for solving cubic equations?
3. How does factoring quadratics relate to finding zeros of functions?
4. What happens if there is no common factor in an equation?
5. How can we check for extraneous solutions in polynomial equations?

Tip:

When factoring cubic equations, always look for a common factor first, as it simplifies the process significantly!