Expanding (x+2)^4 using 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. This theorem is a powerful tool in algebra, allowing us to expand complex expressions without having to multiply them out manually.
Understanding 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) 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 from k = 0 to k = n.
Applying the Binomial Theorem to (x+2)^4
Let's expand (x+2)^4 using the binomial theorem:
-
Identify n: In this case, n = 4.
-
Expand the summation: We need to calculate the terms for k = 0, 1, 2, 3, and 4.
-
Calculate the binomial coefficients:
- (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
-
Substitute and simplify:
(x + 2)^4 = (4 choose 0) * x^4 * 2^0 + (4 choose 1) * x^3 * 2^1 + (4 choose 2) * x^2 * 2^2 + (4 choose 3) * x^1 * 2^3 + (4 choose 4) * x^0 * 2^4
(x + 2)^4 = 1 * x^4 * 1 + 4 * x^3 * 2 + 6 * x^2 * 4 + 4 * x * 8 + 1 * 1 * 16
**(x + 2)^4 = ** x^4 + 8x^3 + 24x^2 + 32x + 16
Conclusion
Therefore, the expanded form of (x+2)^4 using the binomial theorem is x^4 + 8x^3 + 24x^2 + 32x + 16. This demonstrates the power of the binomial theorem in simplifying complex algebraic expressions.