Expand (a + b)^n for Positive Integer n

Expand (a + b)^n for Positive Integer n

Key Concepts

• To expand an expression like (a + b)^n, we generally use the binomial theorem.
• The expansion involves understanding and using the binomial coefficients, which can be found in Pascal’s Triangle. These coefficients can also be calculated using the combination formula.
• Positive integer n refers to the exponent of the binomial.

Steps to Expansion

• Use the formula for the Binomial Theorem, which is (a + b)^n = Σ (nCk) (a^(n-k)) (b^k) from k = 0 to n.
• The term nCk represents the coefficients in the expansion. It can be calculated as n! / (k!(n-k)!).
• Expand the binomial (a + b)^n term by term.
• Possible values of k range from 0 to n. For each term, decrease the power of a by one and increase the power of b by one.

For example

Let’s consider expanding (a + b)^3

• When n=3, we consider the fourth row of Pascal’s Triangle, which is 1, 3, 3, 1.
• Using the formula and coefficients, we get: (a + b)^3 = 1a^3b^0 + 3a^2b^1 + 3a^1b^2 + 1a^0b^3
• Simplifying this gives us: a^3 + 3a^2b + 3ab^2 + b^3.

Practical Tips

• Remember, the sum of the indices of ‘a’ and ‘b’ in each term is equal to the original power ‘n’.
• For larger values of ‘n’, using the binomial theorem along with Pascal’s Triangle or the combination formula is less error-prone than multiplying (a + b) repeatedly.