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(x-1)^4

3 min read Jun 17, 2024
(x-1)^4

Expanding and Simplifying (x - 1)^4

The expression (x - 1)^4 represents a binomial raised to the fourth power. To understand this expression fully, we'll expand it and simplify it.

Understanding the Binomial Theorem

The Binomial Theorem provides a systematic way to expand expressions of the form (x + y)^n. It 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)!).
  • ∑_(k=0)^n indicates the sum from k = 0 to n.

Applying the Binomial Theorem to (x - 1)^4

Let's apply the Binomial Theorem to our expression (x - 1)^4:

  1. Identify n: In this case, n = 4.

  2. Expand the summation: We need to calculate terms for k = 0, 1, 2, 3, and 4.

  3. Calculate each term:

    • k = 0: (4 choose 0) * x^4 * (-1)^0 = 1 * x^4 * 1 = x^4
    • k = 1: (4 choose 1) * x^3 * (-1)^1 = 4 * x^3 * -1 = -4x^3
    • k = 2: (4 choose 2) * x^2 * (-1)^2 = 6 * x^2 * 1 = 6x^2
    • k = 3: (4 choose 3) * x^1 * (-1)^3 = 4 * x * -1 = -4x
    • k = 4: (4 choose 4) * x^0 * (-1)^4 = 1 * 1 * 1 = 1

  4. Combine the terms: (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1

Conclusion

Therefore, the expanded and simplified form of (x - 1)^4 is x^4 - 4x^3 + 6x^2 - 4x + 1. This polynomial represents the result of multiplying (x - 1) by itself four times. Understanding the Binomial Theorem allows us to expand any binomial raised to a power efficiently.

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