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Let's solve the given exercises step by step. The problems ask for calculating the *MCD* (Greatest Common Divisor or GCD) and the *MCM* (Least Common Multiple or LCM) for a set of monomials and polynomials.

### Exercise 1: Calculate the GCD and LCM of the following monomials

#### 1a) \( 4x^2y^3 \) and \( 8x^2y^2z^2 \)

- **GCD**:
  - Coefficients: GCD of \( 4 \) and \( 8 \) is \( 4 \).
  - Variables:
    - \( x^2 \) in both.
    - For \( y \), take the lower power: \( y^2 \).
    - \( z^0 \) in the first term (no \( z \)), so no \( z \) in GCD.
  
  **GCD:** \( 4x^2y^2 \)

- **LCM**:
  - Coefficients: LCM of \( 4 \) and \( 8 \) is \( 8 \).
  - Variables:
    - \( x^2 \) in both.
    - Take the higher power of \( y \): \( y^3 \).
    - Take the higher power of \( z \): \( z^2 \).
  
  **LCM:** \( 8x^2y^3z^2 \)

#### 1b) \( 12x^2yz^3 \), \( 18xyz \), and \( 24x^3yz^2 \)

- **GCD**:
  - Coefficients: GCD of \( 12 \), \( 18 \), and \( 24 \) is \( 6 \).
  - Variables:
    - \( x \) appears in the minimum power as \( x^1 \).
    - \( y \) appears as \( y^1 \) in all terms.
    - \( z \) appears in the minimum power as \( z^1 \).
  
  **GCD:** \( 6xyz \)

- **LCM**:
  - Coefficients: LCM of \( 12 \), \( 18 \), and \( 24 \) is \( 72 \).
  - Variables:
    - Take the highest power of \( x \): \( x^3 \).
    - \( y^1 \) in all.
    - Take the highest power of \( z \): \( z^3 \).
  
  **LCM:** \( 72x^3yz^3 \)

#### 1c) \( 8am^3n \) and \( 20x^2m^2 \)

- **GCD**:
  - Coefficients: GCD of \( 8 \) and \( 20 \) is \( 4 \).
  - Variables:
    - \( m^2 \) (lowest power of \( m \)).
  
  **GCD:** \( 4m^2 \)

- **LCM**:
  - Coefficients: LCM of \( 8 \) and \( 20 \) is \( 40 \).
  - Variables:
    - Highest power of \( m \): \( m^3 \).
  
  **LCM:** \( 40am^3n \)

#### 1d) \( 4x^3y^3 \), \( 8x^2y^2 \), and \( 2x \)

- **GCD**:
  - Coefficients: GCD of \( 4 \), \( 8 \), and \( 2 \) is \( 2 \).
  - Variables:
    - \( x^1 \) (lowest power of \( x \)).
  
  **GCD:** \( 2x \)

- **LCM**:
  - Coefficients: LCM of \( 4 \), \( 8 \), and \( 2 \) is \( 8 \).
  - Variables:
    - Highest power of \( x \): \( x^3 \).
    - Highest power of \( y \): \( y^3 \).
  
  **LCM:** \( 8x^3y^3 \)

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### Exercise 2: Calculate the GCD and LCM of the following polynomials

#### 2a) \( (2a^2 + 2ab) \) and \( (4a^2 - 4ab) \)

- **GCD**:
  - Factor each term: \( 2a(a + b) \) and \( 4a(a - b) \).
  - GCD is \( 2a \) (common factor).
  
  **GCD:** \( 2a \)

- **LCM**:
  - LCM is found by multiplying the GCD by all the unique factors from both polynomials: \( 4a(a + b)(a - b) \).
  
  **LCM:** \( 4a(a + b)(a - b) \)

#### 2b) \( (6x^3y - 6x^2y) \) and \( (9x^3y^2 + 18x^2y^2) \)

- **GCD**:
  - Factor both terms: \( 6x^2y(x - 1) \) and \( 9x^2y^2(x + 2) \).
  - GCD is \( 3x^2y \).
  
  **GCD:** \( 3x^2y \)

- **LCM**:
  - Multiply the GCD by the unique factors: \( 18x^3y^2(x - 1)(x + 2) \).
  
  **LCM:** \( 18x^3y^2(x - 1)(x + 2) \)

#### 2c) \( (a^2 - 1) \), \( (a^2 + 2a + 1) \), and \( (a + 1) \)

- **GCD**:
  - Factor the polynomials: \( (a^2 - 1) = (a - 1)(a + 1) \) and \( (a^2 + 2a + 1) = (a + 1)^2 \).
  - GCD is \( (a + 1) \).
  
  **GCD:** \( (a + 1) \)

- **LCM**:
  - Multiply the GCD by the remaining unique factors: \( (a - 1)(a + 1)^2 \).
  
  **LCM:** \( (a - 1)(a + 1)^2 \)

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Would you like more details on any of the steps or further clarifications?

Here are five related questions to expand the topic:
1. How do you determine the GCD of more than two monomials?
2. What is the difference between factoring monomials and factoring polynomials?
3. Can you apply the Euclidean algorithm to polynomials? If so, how?
4. What is the significance of the LCM in algebraic contexts, beyond simple arithmetic?
5. How would you handle cases where there are no common factors in a set of monomials?

**Tip**: When factoring polynomials, always check for common factors first to simplify the process before dealing with higher powers or multiple terms.

Calcula el MCD y el MCM de los siguientes monomios y polinomios.

Learn how to calculate the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of monomials and polynomials. This guide covers step-by-step solutions for factoring monomials and polynomials to find the GCD and LCM. Suitable for high school students in Grades 9-12.

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