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(m+n)^4

4 min read Jun 16, 2024
(m+n)^4

Expanding (m+n)^4: A Guide Using the Binomial Theorem

The expression (m+n)^4 represents the fourth power of the binomial (m+n). Expanding this expression involves multiplying the binomial by itself four times. While you can do this directly, it becomes tedious for higher powers. The Binomial Theorem offers a much more efficient way to expand such expressions.

Understanding the Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (x+y)^n, where n is a non-negative integer. The theorem states:

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

Where:

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

Expanding (m+n)^4 using the Binomial Theorem

  1. Identify n and the terms: In our case, n = 4, x = m, and y = n.

  2. Calculate the binomial coefficients: We need to calculate the binomial coefficients 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

  3. Apply the Binomial Theorem: Substitute the values into the formula:

    (m + n)^4 = (4 choose 0) * m^4 * n^0 + (4 choose 1) * m^3 * n^1 + (4 choose 2) * m^2 * n^2 + (4 choose 3) * m^1 * n^3 + (4 choose 4) * m^0 * n^4

  4. Simplify the expression:

    (m + n)^4 = m^4 + 4m^3n + 6m^2n^2 + 4mn^3 + n^4

Key Observations

  • The expansion of (m+n)^4 results in a polynomial with 5 terms.
  • The exponents of m decrease from 4 to 0, while the exponents of n increase from 0 to 4.
  • The coefficients follow a symmetrical pattern: 1, 4, 6, 4, 1, which are the elements of Pascal's Triangle for the fourth row.

Conclusion

The Binomial Theorem provides a powerful tool for expanding binomial expressions raised to any power. Applying the theorem to (m+n)^4, we arrive at the expanded form m^4 + 4m^3n + 6m^2n^2 + 4mn^3 + n^4, demonstrating the efficiency and elegance of the theorem.

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