(x+y)^5 Binomial Expansion

5 min read Jun 17, 2024

The Binomial Theorem: Unlocking the Power of (x+y)^5

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (x + y)^n, where n is a non-negative integer. This theorem is particularly useful for expanding expressions with high powers, eliminating the tedious process of multiplying out each term individually.

Understanding the Pattern

The key to understanding the binomial theorem lies in recognizing the patterns that emerge in the expansion of (x + y)^n. Let's look at the expansions for the first few values of n:

(x + y)^0 = 1
(x + y)^1 = x + y
(x + y)^2 = x^2 + 2xy + y^2
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

Notice the following:

The powers of x decrease from n to 0, while the powers of y increase from 0 to n.
The coefficients of each term follow a pattern known as Pascal's Triangle.

Pascal's Triangle: A Visual Aid

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 Pascal's Triangle are:

          1
        1   1
      1   2   1
    1   3   3   1
  1   4   6   4   1

The coefficients of the terms in the binomial expansion correspond to the numbers in each row of Pascal's Triangle.

The Binomial Theorem Formula

The binomial theorem can be expressed as a formula:

(x + y)^n = ∑_(k=0)^n (n choose k) x^(n-k) y^k

where:

(n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!). It represents the number of ways to choose k objects from a set of n objects.
∑_(k=0)^n denotes the sum of all terms from k = 0 to k = n.

Applying the Theorem: (x + y)^5

Let's expand (x + y)^5 using the binomial theorem:

(x + y)^5 = (5 choose 0) x^5 y^0 + (5 choose 1) x^4 y^1 + (5 choose 2) x^3 y^2 + (5 choose 3) x^2 y^3 + (5 choose 4) x^1 y^4 + (5 choose 5) x^0 y^5

Calculating the binomial coefficients:

(5 choose 0) = 1
(5 choose 1) = 5
(5 choose 2) = 10
(5 choose 3) = 10
(5 choose 4) = 5
(5 choose 5) = 1

Substituting the values:

(x + y)^5 = 1x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + 1y^5

Therefore, the expansion of (x + y)^5 is x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5.

Conclusion

The binomial theorem provides a concise and efficient method for expanding expressions of the form (x + y)^n. By understanding the patterns within the expansion and utilizing Pascal's Triangle, we can easily calculate the coefficients and determine the full expanded form. This powerful tool proves invaluable in various mathematical applications, particularly in areas like calculus, probability, and statistics.