(a+b)^12 Binomial Expansion

Jasonbradley

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Jun 16, 2024

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Understanding the Binomial Expansion of (a + b)^12

The binomial theorem provides a powerful formula for expanding expressions of the form (a + b)^n, where 'n' is a non-negative integer. This article focuses on the expansion of (a + b)^12, demonstrating the application of the binomial theorem and its various aspects.

The Binomial Theorem

The binomial theorem states that for any non-negative integer 'n', the following equation holds:

(a + b)^n = ∑_(k=0)^n (n choose k) a^(n-k) b^k

Where:

• (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
• k is an integer ranging from 0 to n.

Expanding (a + b)^12

To expand (a + b)^12, we apply the binomial theorem with n = 12. This results in the following expansion:

(a + b)^12 = ∑_(k=0)^12 (12 choose k) a^(12-k) b^k

Let's break down the expansion term by term:

• k = 0: (12 choose 0) a^12 b^0 = a^12
• k = 1: (12 choose 1) a^11 b^1 = 12a^11 b
• k = 2: (12 choose 2) a^10 b^2 = 66a^10 b^2
• k = 3: (12 choose 3) a^9 b^3 = 220a^9 b^3
• k = 4: (12 choose 4) a^8 b^4 = 495a^8 b^4
• k = 5: (12 choose 5) a^7 b^5 = 792a^7 b^5
• k = 6: (12 choose 6) a^6 b^6 = 924a^6 b^6
• k = 7: (12 choose 7) a^5 b^7 = 792a^5 b^7
• k = 8: (12 choose 8) a^4 b^8 = 495a^4 b^8
• k = 9: (12 choose 9) a^3 b^9 = 220a^3 b^9
• k = 10: (12 choose 10) a^2 b^10 = 66a^2 b^10
• k = 11: (12 choose 11) a^1 b^11 = 12a b^11
• k = 12: (12 choose 12) a^0 b^12 = b^12

Therefore, the complete expansion of (a + b)^12 is:

(a + b)^12 = a^12 + 12a^11 b + 66a^10 b^2 + 220a^9 b^3 + 495a^8 b^4 + 792a^7 b^5 + 924a^6 b^6 + 792a^5 b^7 + 495a^4 b^8 + 220a^3 b^9 + 66a^2 b^10 + 12a b^11 + b^12

Key Observations:

• Symmetry: Notice the symmetrical pattern in the coefficients. The coefficients for terms equidistant from the beginning and end of the expansion are the same.
• Pascal's Triangle: The binomial coefficients can be conveniently obtained from Pascal's Triangle, where each entry is the sum of the two numbers directly above it.
• Applications: The binomial theorem finds wide applications in various fields like probability, statistics, and calculus. It allows us to express complex expressions in a more manageable form.

By understanding the binomial theorem and its application, we can efficiently expand expressions like (a + b)^12 and leverage its power in various mathematical contexts.