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The equation given is:

\[
c^3 - c^2 - 4c + 4 = 0
\]

We need to find the values of \( c \) that satisfy this cubic equation. Let's proceed by factoring or using numerical methods.

### Step 1: Try Rational Root Theorem
The Rational Root Theorem suggests testing possible rational roots by dividing the constant term (4) by the leading coefficient (1). The possible rational roots are factors of 4, which are \( \pm 1, \pm 2, \pm 4 \).

Let's test them:

- For \( c = 1 \):

\[
1^3 - 1^2 - 4(1) + 4 = 1 - 1 - 4 + 4 = 0
\]

So, \( c = 1 \) is a root!

### Step 2: Factor the equation
Now that we know \( c = 1 \) is a root, we can factor \( (c - 1) \) out of the cubic equation. We perform synthetic division or polynomial division to factor \( c^3 - c^2 - 4c + 4 \) by \( (c - 1) \).

\[
\frac{c^3 - c^2 - 4c + 4}{c - 1} = c^2 - 4
\]

### Step 3: Solve the quadratic equation
We now solve the quadratic \( c^2 - 4 = 0 \).

\[
c^2 = 4
\]
\[
c = \pm 2
\]

### Final Answer
The solutions to the equation are:

\[
c = 1, \, c = 2, \, c = -2
\]

Would you like more details on any of the steps? Let me know if you have any questions!

Here are some additional questions to consider:

1. How do we apply the Rational Root Theorem to find potential roots?
2. What are the steps involved in synthetic division?
3. How do we recognize and solve a quadratic equation like \( c^2 - 4 = 0 \)?
4. Can cubic equations always be factored into simpler parts?
5. What are alternative methods to solve cubic equations?

**Tip:** Always test your potential roots by plugging them back into the original equation to verify your solution.

Solve the equation c^3 - c^2 - 4c + 4 = 0.

Learn how to solve the cubic equation c^3 - c^2 - 4c + 4 with a detailed step-by-step solution using factoring and the Rational Root Theorem. Suitable for students in Grades 10-12.

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