(6m+2)^2 Pascal's Triangle

Jasonbradley

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Jun 16, 2024

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Expanding (6m + 2)² using Pascal's Triangle

Pascal's Triangle is a powerful tool for expanding binomials, especially when dealing with higher powers. Let's explore how it helps us expand (6m + 2)².

Understanding Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The first few rows of the triangle look like this:

        1
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1
   1 5 10 10 5 1

The rows of the triangle correspond to the power of the binomial we are expanding. For example, the 3rd row helps us expand (a + b)², the 4th row helps us expand (a + b)³, and so on.

Expanding (6m + 2)²

To expand (6m + 2)² using Pascal's Triangle, we need to consider the second row of the triangle, which is: 1 2 1. These numbers represent the coefficients for each term in our expansion.

1. Identify the terms: Our binomial is (6m + 2). Let's call 6m 'a' and 2 'b'.

2. Apply the coefficients:

   • The first term is 1 * a² * b⁰ = 1 * (6m)² * 2⁰ = 36m²
   • The second term is 2 * a¹ * b¹ = 2 * (6m)¹ * 2¹ = 24m
   • The third term is 1 * a⁰ * b² = 1 * (6m)⁰ * 2² = 4

3. Combine the terms: Adding the terms together gives us the expanded form: (6m + 2)² = 36m² + 24m + 4

Conclusion

Using Pascal's Triangle simplifies the process of expanding binomials. By understanding the pattern and applying the coefficients, we can efficiently expand expressions like (6m + 2)² without having to perform lengthy multiplication.