%5E6-expand.jpeg "(a+b)^6 Expand")
Expanding (a + b)^6: The Binomial Theorem
The binomial theorem is a powerful tool for expanding expressions of the form (a + b)^n, where n is a non-negative integer. This theorem allows us to efficiently determine the coefficients of the terms in the expansion.
Understanding the Binomial Theorem
The binomial theorem states that:
(a + b)^n = ∑(n choose k) * a^(n-k) * b^k,
where:
- n is the power of the binomial.
- k ranges from 0 to n.
- (n choose k) represents the binomial coefficient, which is calculated as n!/(k!*(n-k)!).
Expanding (a + b)^6
To expand (a + b)^6, we can use the binomial theorem by setting n = 6 and applying the formula:
(a + b)^6 = ∑(6 choose k) * a^(6-k) * b^k
Let's calculate each term for k from 0 to 6:
- k = 0: (6 choose 0) * a^(6-0) * b^0 = 1 * a^6 * 1 = a^6
- k = 1: (6 choose 1) * a^(6-1) * b^1 = 6 * a^5 * b = 6a^5b
- k = 2: (6 choose 2) * a^(6-2) * b^2 = 15 * a^4 * b^2 = 15a^4b^2
- k = 3: (6 choose 3) * a^(6-3) * b^3 = 20 * a^3 * b^3 = 20a^3b^3
- k = 4: (6 choose 4) * a^(6-4) * b^4 = 15 * a^2 * b^4 = 15a^2b^4
- k = 5: (6 choose 5) * a^(6-5) * b^5 = 6 * a^1 * b^5 = 6ab^5
- k = 6: (6 choose 6) * a^(6-6) * b^6 = 1 * a^0 * b^6 = b^6
Therefore, the complete expansion of (a + b)^6 is:
(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6
Key Observations
- Symmetry: Notice the symmetry in the coefficients. The coefficients are mirrored from the beginning and the end of the expansion.
- Pascal's Triangle: The coefficients of the binomial expansion can be easily found using Pascal's Triangle.
- Applications: The binomial theorem has extensive applications in algebra, calculus, probability, and statistics.
By applying the binomial theorem, we can efficiently expand any expression of the form (a + b)^n, making complex calculations much simpler.