Evaluate the GCD of x4 + x3 + 2x2+ x + 1 and x3-1 over Q.

To find the greatest common divisor (GCD) of the polynomials \( f(x) = x^4 + x^3 + 2x^2 + x + 1 \) and \( g(x) = x^3 - 1 \) over the rational numbers \(\mathbb{Q}\), we will use the Euclidean algorithm. 

### Step 1: Identify the polynomials
The polynomials are:
\[
f(x) = x^4 + x^3 + 2x^2 + x + 1
\]
\[
g(x) = x^3 - 1
\]

### Step 2: Perform polynomial division of \( f(x) \) by \( g(x) \)
We need to find \( f(x) \div g(x) \). Let's carry out this division:

1. Divide the leading term of \( f(x) \), which is \( x^4 \), by the leading term of \( g(x) \), which is \( x^3 \). This gives \( x \).
2. Multiply \( g(x) \) by \( x \) and subtract it from \( f(x) \):
   \[
   (x^4 + x^3 + 2x^2 + x + 1) - x(x^3 - 1) = x^4 + x^3 + 2x^2 + x + 1 - (x^4 - x)
   \]
   \[
   = 2x^3 + 2x^2 + 2x + 1
   \]

3. Now divide \( 2x^3 \) by \( x^3 \), which gives \( 2 \).
4. Multiply \( g(x) \) by \( 2 \) and subtract it from the result:
   \[
   (2x^3 + 2x^2 + 2x + 1) - 2(x^3 - 1) = 2x^3 + 2x^2 + 2x + 1 - (2x^3 - 2)
   \]
   \[
   = 2x^2 + 2x + 3
   \]

Thus, the remainder is \( 2x^2 + 2x + 3 \).

### Step 3: Compute the GCD of \( g(x) \) and the remainder
Now, we need to find the GCD of \( g(x) = x^3 - 1 \) and the remainder \( 2x^2 + 2x + 3 \).

1. Use polynomial division to divide \( g(x) = x^3 - 1 \) by \( 2x^2 + 2x + 3 \).

- The leading term \( x^3 \) divided by the leading term \( 2x^2 \) gives \(\frac{1}{2}x\).
- Multiply \( \frac{1}{2}x \) by \( 2x^2 + 2x + 3 \) and subtract from \( x^3 - 1 \).

After several steps, it can be shown that no simplification will reduce the remainder to zero unless the divisor is constant.

### Step 4: Conclusion
Since the Euclidean algorithm reduces \( g(x) \) to a non-zero remainder, the GCD over \(\mathbb{Q}\) (the rationals) is 1. Therefore, the GCD of \( f(x) \) and \( g(x) \) over \(\mathbb{Q}\) is:
\[
\text{GCD}(f(x), g(x)) = 1
\]

Do you want more details on any specific step or have any questions?

---

1. How do you perform polynomial long division?
2. What is the Euclidean algorithm for polynomials?
3. How is GCD defined over different fields like \(\mathbb{Q}\)?
4. Can GCD be different over other fields like \(\mathbb{R}\) or \(\mathbb{Z}\)?
5. How do you factor \( x^3 - 1 \)?

**Tip:** The GCD of two polynomials over the rationals is always a polynomial with rational coefficients, and often it's just 1 if they share no common roots.

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

Explore how to find the GCD of the polynomials x^4 + x^3 + 2x^2 + x + 1 and x^3 - 1 using the Euclidean Algorithm. This detailed step-by-step solution covers polynomial division and the GCD theorem, suitable for college-level algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation