%5E6-binomial-expansion.jpeg "(2x-y)^6 Binomial Expansion")
Binomial Expansion of (2x - y)^6
The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. In this case, we'll apply it to the expression (2x - y)^6.
The Binomial Theorem
The binomial theorem states:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
where:
- Σ represents the summation from k = 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
- a and b are the terms in the binomial
- n is the power to which the binomial is raised
Expanding (2x - y)^6
Let's apply the binomial theorem to our expression:
(2x - y)^6 = Σ (6 choose k) * (2x)^(6-k) * (-y)^k
Now, we can expand this summation for each value of k from 0 to 6:
- k = 0: (6 choose 0) * (2x)^6 * (-y)^0 = 1 * 64x^6 * 1 = 64x^6
- k = 1: (6 choose 1) * (2x)^5 * (-y)^1 = 6 * 32x^5 * -y = -192x^5y
- k = 2: (6 choose 2) * (2x)^4 * (-y)^2 = 15 * 16x^4 * y^2 = 240x^4y^2
- k = 3: (6 choose 3) * (2x)^3 * (-y)^3 = 20 * 8x^3 * -y^3 = -160x^3y^3
- k = 4: (6 choose 4) * (2x)^2 * (-y)^4 = 15 * 4x^2 * y^4 = 60x^2y^4
- k = 5: (6 choose 5) * (2x)^1 * (-y)^5 = 6 * 2x * -y^5 = -12xy^5
- k = 6: (6 choose 6) * (2x)^0 * (-y)^6 = 1 * 1 * y^6 = y^6
Therefore, the complete expansion of (2x - y)^6 is:
(2x - y)^6 = 64x^6 - 192x^5y + 240x^4y^2 - 160x^3y^3 + 60x^2y^4 - 12xy^5 + y^6
Conclusion
By applying the binomial theorem, we successfully expanded the expression (2x - y)^6. This demonstrates the power of the theorem in simplifying complex algebraic expressions.