%5E4-expanded.jpeg "(2+x)^4 Expanded")
Expanding (2+x)^4
The expression (2+x)^4 can be expanded using the Binomial Theorem. This theorem provides a formula to expand any binomial raised to a power.
The Binomial Theorem
The Binomial Theorem states:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
where:
- n is the power to which the binomial is raised
- k ranges from 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
Expanding (2+x)^4
Applying the Binomial Theorem, we get:
(2+x)^4 = Σ (4 choose k) * 2^(4-k) * x^k
Expanding the summation for k from 0 to 4:
- k = 0: (4 choose 0) * 2^(4-0) * x^0 = 1 * 16 * 1 = 16
- k = 1: (4 choose 1) * 2^(4-1) * x^1 = 4 * 8 * x = 32x
- k = 2: (4 choose 2) * 2^(4-2) * x^2 = 6 * 4 * x^2 = 24x^2
- k = 3: (4 choose 3) * 2^(4-3) * x^3 = 4 * 2 * x^3 = 8x^3
- k = 4: (4 choose 4) * 2^(4-4) * x^4 = 1 * 1 * x^4 = x^4
Therefore, the expanded form of (2+x)^4 is:
(2+x)^4 = 16 + 32x + 24x^2 + 8x^3 + x^4
Conclusion
By applying the Binomial Theorem, we can systematically expand expressions like (2+x)^4. This method provides a clear and concise way to calculate all the terms involved in the expansion.