%5E6-expand.jpeg "(x-2)^6 Expand")
Expanding (x-2)^6 using the Binomial Theorem
The binomial theorem provides a systematic way to expand expressions of the form (x + y)^n. Let's apply it to expand (x - 2)^6.
The Binomial Theorem
The binomial theorem states:
(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)!).
- ∑_(k=0)^n indicates the sum from k = 0 to n.
Expanding (x - 2)^6
Let's break down the expansion step by step:
-
Identify n and y: In our case, n = 6 and y = -2.
-
Calculate the binomial coefficients:
- (6 choose 0) = 6! / (0! * 6!) = 1
- (6 choose 1) = 6! / (1! * 5!) = 6
- (6 choose 2) = 6! / (2! * 4!) = 15
- (6 choose 3) = 6! / (3! * 3!) = 20
- (6 choose 4) = 6! / (4! * 2!) = 15
- (6 choose 5) = 6! / (5! * 1!) = 6
- (6 choose 6) = 6! / (6! * 0!) = 1
-
Apply the binomial theorem:
(x - 2)^6 = (6 choose 0) x^6 (-2)^0 + (6 choose 1) x^5 (-2)^1 + (6 choose 2) x^4 (-2)^2 + (6 choose 3) x^3 (-2)^3 + (6 choose 4) x^2 (-2)^4 + (6 choose 5) x^1 (-2)^5 + (6 choose 6) x^0 (-2)^6
-
Simplify:
(x - 2)^6 = 1x^6 + 6x^5 (-2) + 15x^4 (4) + 20x^3 (-8) + 15x^2 (16) + 6x (-32) + 1 (64)
-
Final expansion:
(x - 2)^6 = x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
Conclusion
By applying the binomial theorem, we have successfully expanded (x - 2)^6. This method provides a structured approach to expanding binomial expressions raised to any power.