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We are tasked with finding the greatest common divisor (GCD) of the two polynomials:

1. \( f(x) = x^4 + x^3 + 2x^2 + x + 1 \)
2. \( g(x) = x^3 - 1 \)

over the field of rational numbers \( \mathbb{Q} \).

### Step-by-step solution:

#### 1. Factor \( g(x) = x^3 - 1 \):
We know that \( x^3 - 1 \) can be factored as:
\[
x^3 - 1 = (x - 1)(x^2 + x + 1)
\]
Thus, we rewrite \( g(x) \) as:
\[
g(x) = (x - 1)(x^2 + x + 1)
\]

#### 2. Check if any factors divide \( f(x) \):
Next, we need to determine if any of the factors of \( g(x) \), specifically \( x - 1 \) and \( x^2 + x + 1 \), divide \( f(x) = x^4 + x^3 + 2x^2 + x + 1 \).

We will perform polynomial division to see if either \( x - 1 \) or \( x^2 + x + 1 \) divides \( f(x) \).

#### 3. Divide \( f(x) \) by \( x - 1 \):
Using synthetic or long division, we divide \( f(x) \) by \( x - 1 \). 

Without going into the detailed steps of the division, it turns out that the division does not result in a remainder of zero, meaning \( x - 1 \) is **not a divisor** of \( f(x) \).

#### 4. Divide \( f(x) \) by \( x^2 + x + 1 \):
Now, we divide \( f(x) \) by \( x^2 + x + 1 \). 

After performing the division, we find that:
\[
f(x) = (x^2 + x + 1)(x^2 + 1)
\]
This shows that \( x^2 + x + 1 \) is indeed a divisor of \( f(x) \).

#### 5. GCD conclusion:
Since \( x^2 + x + 1 \) divides both \( f(x) \) and \( g(x) \), the GCD of \( f(x) \) and \( g(x) \) over \( \mathbb{Q} \) is:
\[
\text{GCD}(f(x), g(x)) = x^2 + x + 1
\]

Would you like more details or clarifications on any step?

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Here are some related questions to further explore this topic:

1. How do you perform polynomial division step-by-step for these types of problems?
2. What are the general methods for finding the GCD of polynomials over a field?
3. Can you factor higher-degree polynomials similarly? What techniques would be used?
4. How does the GCD of polynomials differ when working over fields like \( \mathbb{R} \) or \( \mathbb{C} \)?
5. What is the significance of factoring polynomials in algebraic structures?

**Tip:** When working with polynomials, recognizing common factorizations (like \( x^3 - 1 \)) can simplify your calculations significantly.

Evaluate the GCD of x^4 + x^3 + 2x^2 + x + 1 and x^3 - 1 over Q.

Learn how to evaluate the greatest common divisor (GCD) of the polynomials x^4 + x^3 + 2x^2 + x + 1 and x^3 - 1 over the rational field Q. Explore factoring methods and polynomial division for a step-by-step solution. Suitable for advanced high school and undergraduate students.

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