https://file.aiquickdraw.com/mathbot/file/1725415082375tkbehstg.jpg

The polynomial provided in the image is:

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

To find the rational roots of this polynomial, we can use the **Rational Root Theorem**. The theorem states that any rational root, expressed as \(\frac{p}{q}\), must be such that:
- \(p\) is a factor of the constant term (the term without \(x\)), which in this case is \(-4\).
- \(q\) is a factor of the leading coefficient (the coefficient of the highest power of \(x\)), which in this case is \(1\).

### Step 1: Identify factors of the constant term and leading coefficient.
- The constant term is \(-4\), and its factors are \(\pm 1, \pm 2, \pm 4\).
- The leading coefficient is \(1\), and its factors are \(\pm 1\).

### Step 2: Possible rational roots.
The possible rational roots are:

\[
\pm 1, \pm 2, \pm 4
\]

### Step 3: Testing the possible roots.
We substitute each possible root into the polynomial \(P(x)\) to see if it yields zero.

1. **Test \(x = 1\):**
   \[
   P(1) = (1)^4 - 2(1)^3 + 2(1) - 4 = 1 - 2 + 2 - 4 = -3 \quad (\text{Not a root})
   \]

2. **Test \(x = -1\):**
   \[
   P(-1) = (-1)^4 - 2(-1)^3 + 2(-1) - 4 = 1 + 2 - 2 - 4 = -3 \quad (\text{Not a root})
   \]

3. **Test \(x = 2\):**
   \[
   P(2) = (2)^4 - 2(2)^3 + 2(2) - 4 = 16 - 16 + 4 - 4 = 0 \quad (\text{This is a root!})
   \]

4. **Test \(x = -2\):**
   \[
   P(-2) = (-2)^4 - 2(-2)^3 + 2(-2) - 4 = 16 + 16 - 4 - 4 = 24 \quad (\text{Not a root})
   \]

5. **Test \(x = 4\):**
   \[
   P(4) = (4)^4 - 2(4)^3 + 2(4) - 4 = 256 - 128 + 8 - 4 = 132 \quad (\text{Not a root})
   \]

6. **Test \(x = -4\):**
   \[
   P(-4) = (-4)^4 - 2(-4)^3 + 2(-4) - 4 = 256 + 128 - 8 - 4 = 372 \quad (\text{Not a root})
   \]

### Conclusion:
The only rational root is \(x = 2\).

Would you like further details on how to factor the polynomial after finding the root, or do you have any questions? Here are some related questions to consider:

1. How do you factor a polynomial after finding one of its roots?
2. What is synthetic division, and how is it used with polynomials?
3. How can we determine if a polynomial has irrational or complex roots?
4. What is the difference between a rational root and an irrational root?
5. How do you use the Rational Root Theorem for polynomials with leading coefficients other than 1?

**Tip:** When testing possible roots, always start with the smallest values and use substitution to quickly identify roots.

Find the rational roots of P(x) = x^4 - 2x^3 + 2x - 4.

Find the rational roots of the polynomial equation P(x) = x^4 - 2x^3 + 2x - 4 using the Rational Root Theorem. Explore the detailed solution involving factors of the constant and leading coefficients, suitable for students in Grades 10-12.

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