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The document contains polynomial factoring problems that require the application of the Rational Root Theorem and other factoring techniques. Let's begin by solving the first one as an example.

### Problem 1:
\[ 2x^4 - x^3 - 9x^2 + 4x + 4 \]

#### Step 1: Apply the Rational Root Theorem
- The Rational Root Theorem helps identify possible rational roots using factors of the constant term (4) and the leading coefficient (2).
- Possible roots are: \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2} \).

#### Step 2: Test the possible rational roots
Use synthetic division or substitution to check the possible roots. After testing, we find that \( x = -1 \) is a root.

#### Step 3: Perform synthetic division by \( x + 1 \)
Dividing the polynomial \( 2x^4 - x^3 - 9x^2 + 4x + 4 \) by \( x + 1 \) gives the quotient:
\[ 2x^3 - 3x^2 - 6x + 4 \]

#### Step 4: Factor the quotient further
Now, we factor \( 2x^3 - 3x^2 - 6x + 4 \). Using synthetic division or factoring by grouping, we find another root at \( x = 2 \).

Dividing \( 2x^3 - 3x^2 - 6x + 4 \) by \( x - 2 \) results in:
\[ 2x^2 + x - 2 \]

#### Step 5: Factor the quadratic
Factor \( 2x^2 + x - 2 \):
\[ 2x^2 + x - 2 = (2x - 1)(x + 2) \]

#### Step 6: Final factorization
Thus, the complete factorization of \( 2x^4 - x^3 - 9x^2 + 4x + 4 \) is:
\[
(x + 1)(x - 2)(2x - 1)(x + 2)
\]

Would you like me to proceed with solving the other problems? 

Here are five questions related to this factorization process:
1. What are the steps involved in synthetic division?
2. How do you apply the Rational Root Theorem to find possible roots?
3. Can a polynomial have non-rational roots when using the Rational Root Theorem?
4. What is factoring by grouping, and when is it used?
5. How can you verify that a factorization is correct?

**Tip:** Always check the results by multiplying the factors back to confirm they match the original polynomial.

Factor the following polynomials completely using the rational root theorem and other strategies of your choice.

Practice polynomial factoring using the Rational Root Theorem and synthetic division. Problems include high-degree polynomials with detailed solutions. Suitable for high school students in Grades 9-12.

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