%5E4-expand-formula.jpeg "(x-1)^4 Expand Formula")
Expanding (x-1)^4: A Step-by-Step Guide
The expression (x-1)^4 represents the product of (x-1) multiplied by itself four times: (x-1) * (x-1) * (x-1) * (x-1). Expanding this expression can be done using the binomial theorem or by repeated multiplication.
Using the Binomial Theorem
The binomial theorem provides a formula to expand expressions of the form (a + b)^n. In our case, a = x, b = -1, and n = 4. The formula is:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
where (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying this to our problem:
(x - 1)^4 = Σ (4 choose k) * x^(4-k) * (-1)^k
We need to expand the formula for k = 0, 1, 2, 3, and 4:
- k = 0: (4 choose 0) * x^(4-0) * (-1)^0 = 1 * x^4 * 1 = x^4
- k = 1: (4 choose 1) * x^(4-1) * (-1)^1 = 4 * x^3 * -1 = -4x^3
- k = 2: (4 choose 2) * x^(4-2) * (-1)^2 = 6 * x^2 * 1 = 6x^2
- k = 3: (4 choose 3) * x^(4-3) * (-1)^3 = 4 * x^1 * -1 = -4x
- k = 4: (4 choose 4) * x^(4-4) * (-1)^4 = 1 * x^0 * 1 = 1
Therefore, the expanded form of (x-1)^4 is:
(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
Expanding by Repeated Multiplication
Alternatively, we can expand (x-1)^4 by repeated multiplication:
- (x-1) * (x-1) = x^2 - 2x + 1
- (x^2 - 2x + 1) * (x - 1) = x^3 - 3x^2 + 3x - 1
- (x^3 - 3x^2 + 3x - 1) * (x - 1) = x^4 - 4x^3 + 6x^2 - 4x + 1
Both methods lead to the same result: (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1.
Conclusion
Expanding (x-1)^4 can be achieved using the binomial theorem or by repeated multiplication. The binomial theorem offers a more general and efficient approach, especially for larger exponents. Understanding the expansion of this expression is essential for various mathematical applications, including calculus and algebra.