%5E4-binomial-expansion.jpeg "(1+x)^4 Binomial Expansion")
Understanding the Binomial Expansion of (1+x)^4
The binomial theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. Let's focus on the specific case of (1+x)^4.
The Binomial Theorem
The binomial theorem states that:
(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 choose k) = n! / (k! * (n-k)!)
Expanding (1+x)^4
Applying the binomial theorem to (1+x)^4, we get:
(1 + x)^4 = (4 choose 0) * 1^4 * x^0 + (4 choose 1) * 1^3 * x^1 + (4 choose 2) * 1^2 * x^2 + (4 choose 3) * 1^1 * x^3 + (4 choose 4) * 1^0 * x^4
Let's 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
Now, substituting these values back into the expansion:
(1 + x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4
Summary
Therefore, the binomial expansion of (1+x)^4 is 1 + 4x + 6x^2 + 4x^3 + x^4. This expansion provides a clear understanding of how the powers of x and the constant term combine to form the complete expression.