%5E6-binomial-expansion.jpeg "(2x-y)^6 Binomial Expansion")
Binomial Expansion of (2x-y)^6
In algebra, the binomial theorem is a powerful tool for expanding expressions of the form (a+b)^n, where a and b are variables and n is a positive integer. In this article, we will explore the binomial expansion of (2x-y)^6.
Binomial Expansion Formula
The binomial expansion formula is given by:
(a+b)^n = ∑[n choose k] * a^(n-k) * b^k
where n is a positive integer, and k ranges from 0 to n.
Expanding (2x-y)^6
To expand (2x-y)^6, we will use the binomial expansion formula with a = 2x and b = -y.
(2x-y)^6 = ∑[6 choose k] * (2x)^(6-k) * (-y)^k
Calculating the Expansion
Using the formula, we can calculate the expansion of (2x-y)^6 as follows:
k = 0
[6 choose 0] * (2x)^(6-0) * (-y)^0 = 1 * (2x)^6 * 1 = 64x^6
k = 1
[6 choose 1] * (2x)^(6-1) * (-y)^1 = 6 * (2x)^5 * -y = -192x^5y
k = 2
[6 choose 2] * (2x)^(6-2) * (-y)^2 = 15 * (2x)^4 * y^2 = 240x^4y^2
k = 3
[6 choose 3] * (2x)^(6-3) * (-y)^3 = 20 * (2x)^3 * -y^3 = -160x^3y^3
k = 4
[6 choose 4] * (2x)^(6-4) * (-y)^4 = 15 * (2x)^2 * y^4 = 120x^2y^4
k = 5
[6 choose 5] * (2x)^(6-5) * (-y)^5 = 6 * 2x * -y^5 = -48xy^5
k = 6
[6 choose 6] * (2x)^(6-6) * (-y)^6 = 1 * 1 * y^6 = y^6
Final Expansion
The final expansion of (2x-y)^6 is:
(2x-y)^6 = 64x^6 - 192x^5y + 240x^4y^2 - 160x^3y^3 + 120x^2y^4 - 48xy^5 + y^6
This is the binomial expansion of (2x-y)^6. The expansion contains 7 terms, with coefficients that follow the binomial coefficient pattern.
