To find the zeros of the function \( f(x) = x^3 - 5x^2 + 6x \), we need to solve the equation \( f(x) = 0 \).

### Step 1: Set up the equation
\[
x^3 - 5x^2 + 6x = 0
\]

### Step 2: Factor out \( x \)
Since each term contains \( x \), we can factor out \( x \) from the expression:
\[
x(x^2 - 5x + 6) = 0
\]

This gives us two factors:
1. \( x = 0 \)
2. \( x^2 - 5x + 6 = 0 \)

### Step 3: Solve \( x^2 - 5x + 6 = 0 \)
To solve \( x^2 - 5x + 6 = 0 \), we factor it further:
\[
x^2 - 5x + 6 = (x - 2)(x - 3) = 0
\]

### Step 4: Find the solutions
Setting each factor equal to zero gives us:
1. \( x = 0 \)
2. \( x - 2 = 0 \Rightarrow x = 2 \)
3. \( x - 3 = 0 \Rightarrow x = 3 \)

### Solution
The zeros of \( f(x) = x^3 - 5x^2 + 6x \) are:
\[
x = 0, \quad x = 2, \quad x = 3
\]

Would you like a detailed explanation of factoring or have any other questions?

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### Related Questions:
1. How do you factor trinomials in quadratic form?
2. What are the zeros of the polynomial \( g(x) = x^3 - 4x^2 + 4x \)?
3. How can you verify the zeros of a polynomial?
4. What is the significance of multiplicity in the zeros of a polynomial?
5. How do the zeros of a polynomial relate to its graph?

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**Tip:** When finding zeros, always start by factoring out any common terms.

Find the zeros of f(x) = x^3 − 5x^2 + 6x

Discover how to find the zeros of the polynomial function f(x) = x^3 - 5x^2 + 6x through step-by-step factoring techniques. This example illustrates key algebraic concepts, such as factoring and using the Zero Product Property, and is ideal for students in Grades 9-12.

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