Question
Find the binomial expansion of (2x+y)^4
Answers
GPT 3.5
To find the binomial expansion of (2x + y)^4, we can use the Binomial Theorem.
The Binomial Theorem states that for any positive integer n:
(x + y)^n = C(n, 0) * x^n * y^0 + C(n, 1) * x^(n-1) * y^1 + C(n, 2) * x^(n-2) * y^2 + ... + C(n, n-1) * x^1 * y^(n-1) + C(n, n) * x^0 * y^n
where C(n, k) represents the binomial coefficient, defined as C(n, k) = n! / (k!(n-k)!). Therefore, we can find the binomial expansion of (2x + y)^4 as follows:
(2x + y)^4 = C(4, 0) * (2x)^4 * y^0 + C(4, 1) * (2x)^3 * y^1 + C(4, 2) * (2x)^2 * y^2 + C(4, 3) * (2x)^1 * y^3 + C(4, 4) * (2x)^0 * y^4
Simplifying each term, we get:
(2x + y)^4 = 1 * 16x^4 * 1 + 4 * 8x^3 * y + 6 * 4x^2 * y^2 + 4 * 2x * y^3 + 1 * 1 * y^4
(2x + y)^4 = 16x^4 + 32x^3y + 24x^2y^2 + 8xy^3 + y^4
Therefore, the binomial expansion of (2x + y)^4 is:
16x^4 + 32x^3y + 24x^2y^2 + 8xy^3 + y^4
The Binomial Theorem states that for any positive integer n:
(x + y)^n = C(n, 0) * x^n * y^0 + C(n, 1) * x^(n-1) * y^1 + C(n, 2) * x^(n-2) * y^2 + ... + C(n, n-1) * x^1 * y^(n-1) + C(n, n) * x^0 * y^n
where C(n, k) represents the binomial coefficient, defined as C(n, k) = n! / (k!(n-k)!). Therefore, we can find the binomial expansion of (2x + y)^4 as follows:
(2x + y)^4 = C(4, 0) * (2x)^4 * y^0 + C(4, 1) * (2x)^3 * y^1 + C(4, 2) * (2x)^2 * y^2 + C(4, 3) * (2x)^1 * y^3 + C(4, 4) * (2x)^0 * y^4
Simplifying each term, we get:
(2x + y)^4 = 1 * 16x^4 * 1 + 4 * 8x^3 * y + 6 * 4x^2 * y^2 + 4 * 2x * y^3 + 1 * 1 * y^4
(2x + y)^4 = 16x^4 + 32x^3y + 24x^2y^2 + 8xy^3 + y^4
Therefore, the binomial expansion of (2x + y)^4 is:
16x^4 + 32x^3y + 24x^2y^2 + 8xy^3 + y^4