(x+y)^4 Expand Formula

4 min read Jun 17, 2024

Expanding (x + y)^4: A Comprehensive Guide

The expansion of (x + y)^4 is a common topic in algebra, particularly when dealing with binomial theorem and algebraic manipulations. Let's explore how to expand this expression and understand the underlying principles.

The Binomial Theorem

The binomial theorem provides a systematic way to expand expressions of the form (x + y)^n for any positive integer n. The general formula is:

(x + y)^n = ∑ (n choose k) x^(n-k) y^k, where k goes from 0 to n

Here, (n choose k) represents the binomial coefficient, which can be calculated as n!/(k!(n-k)!).

Expanding (x + y)^4

Let's apply the binomial theorem to expand (x + y)^4:

Identify n: In this case, n = 4.
Calculate the binomial coefficients: We need to calculate (4 choose k) for k = 0, 1, 2, 3, and 4.
- (4 choose 0) = 4!/(0!4!) = 1
- (4 choose 1) = 4!/(1!3!) = 4
- (4 choose 2) = 4!/(2!2!) = 6
- (4 choose 3) = 4!/(3!1!) = 4
- (4 choose 4) = 4!/(4!0!) = 1
Apply the formula:
- (x + y)^4 = (4 choose 0) x^4 y^0 + (4 choose 1) x^3 y^1 + (4 choose 2) x^2 y^2 + (4 choose 3) x^1 y^3 + (4 choose 4) x^0 y^4
Simplify:
- (x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

Understanding the Pattern

Notice the pattern in the expansion of (x + y)^4:

The powers of x decrease: Starting from x^4, the power of x decreases by 1 in each term.
The powers of y increase: Starting from y^0, the power of y increases by 1 in each term.
The coefficients follow Pascal's Triangle: The binomial coefficients (1, 4, 6, 4, 1) correspond to the fourth row of Pascal's Triangle.

Applications of (x + y)^4 Expansion

The expansion of (x + y)^4 has numerous applications in various fields, including:

Algebraic manipulations: Simplifying expressions and solving equations.
Calculus: Finding derivatives and integrals of functions involving binomials.
Statistics: Probability calculations and modeling.

Conclusion

Expanding (x + y)^4 using the binomial theorem is a powerful tool for understanding and manipulating algebraic expressions. By applying the formula and recognizing the pattern, you can efficiently expand any binomial raised to a positive integer power.