To factor the cubic expression \( x^3 - 4x^2 + 4x - 3 \), let's start by testing potential rational roots using the Rational Root Theorem. The possible rational roots are factors of the constant term (-3) divided by factors of the leading coefficient (1), which are:

\[
\pm 1, \pm 3
\]

Let's test these possible roots:

1. **Test \( x = 1 \):**
\[
1^3 - 4(1^2) + 4(1) - 3 = 1 - 4 + 4 - 3 = -2
\]
This is not a root.

2. **Test \( x = -1 \):**
\[
(-1)^3 - 4(-1)^2 + 4(-1) - 3 = -1 - 4 - 4 - 3 = -12
\]
This is not a root.

3. **Test \( x = 3 \):**
\[
3^3 - 4(3^2) + 4(3) - 3 = 27 - 36 + 12 - 3 = 0
\]
So, \( x = 3 \) is a root.

Now that we know \( x = 3 \) is a root, we can factor \( (x - 3) \) from the cubic expression. We can do this using synthetic division or long division.

### Synthetic Division of \( x^3 - 4x^2 + 4x - 3 \) by \( x - 3 \):

```
3 |   1    -4    4   -3
   |        3   -3    3
   ---------------------
     1    -1    1    0
```

The quotient is \( x^2 - x + 1 \), so we can write the factored form as:

\[
x^3 - 4x^2 + 4x - 3 = (x - 3)(x^2 - x + 1)
\]

This is the fully factored form of the cubic expression.

Would you like further details or clarification on any of these steps?

Here are five related questions to deepen your understanding:
1. How do you apply the Rational Root Theorem in other polynomial problems?
2. What are some methods for factoring cubic equations?
3. How do you perform synthetic division step-by-step?
4. Can all cubic polynomials be factored using rational roots?
5. What are some characteristics of the quadratic factor \(x^2 - x + 1\)?

**Tip:** Always test possible roots systematically when using the Rational Root Theorem—it saves time in factoring!

Discover how to factor the cubic expression x³ - 4x² + 4x - 3 using the Rational Root Theorem and synthetic division. This step-by-step solution explores algebraic concepts suitable for high school students in Grades 10-12.

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