%5E4-binomial-theorem.jpeg "(2x+y)^4 Binomial Theorem")
Expanding (2x + y)^4 using the Binomial Theorem
The binomial theorem provides a powerful tool for expanding expressions of the form (a + b)^n, where 'a' and 'b' are any real numbers and 'n' is a non-negative integer. Let's apply this theorem to expand (2x + y)^4.
The Binomial Theorem
The binomial theorem states:
(a + b)^n = ∑_(k=0)^n (n_C_k) a^(n-k) b^k
Where:
- nCk represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
- ∑_(k=0)^n denotes the sum from k = 0 to k = n.
Expanding (2x + y)^4
Using the binomial theorem, we can expand (2x + y)^4 as follows:
(2x + y)^4 = ∑_(k=0)^4 (4_C_k) (2x)^(4-k) y^k
Let's expand this expression term by term:
- k = 0: (4_C_0) (2x)^4 y^0 = 1 * 16x^4 * 1 = 16x^4
- k = 1: (4_C_1) (2x)^3 y^1 = 4 * 8x^3 * y = 32x^3y
- k = 2: (4_C_2) (2x)^2 y^2 = 6 * 4x^2 * y^2 = 24x^2y^2
- k = 3: (4_C_3) (2x)^1 y^3 = 4 * 2x * y^3 = 8xy^3
- k = 4: (4_C_4) (2x)^0 y^4 = 1 * 1 * y^4 = y^4
Therefore, the expanded form of (2x + y)^4 is:
(2x + y)^4 = 16x^4 + 32x^3y + 24x^2y^2 + 8xy^3 + y^4
Key Points
- Understanding the Binomial Coefficient: The binomial coefficient represents the number of ways to choose k items from a set of n items, without regard to order. In our example, 4_C_1 = 4, meaning there are 4 ways to choose 1 item from a set of 4 items.
- Applying the Theorem: The binomial theorem provides a systematic way to expand any binomial raised to a non-negative integer power. It simplifies the process of multiplying out the terms.
- Pattern Recognition: Notice the pattern in the coefficients and exponents of the expanded terms. The coefficients follow Pascal's Triangle, and the exponents of 'x' decrease while the exponents of 'y' increase with each term.