%5E5-binomial-expansion.jpeg "(2x-3y)^5 Binomial Expansion")
Binomial Expansion of (2x-3y)^5
The binomial theorem is a powerful tool for expanding expressions of the form (a + b)^n, where n is a positive integer. In this case, we'll use it to expand (2x - 3y)^5.
Understanding the Binomial Theorem
The binomial theorem states that:
(a + b)^n = ∑(n choose k) * a^(n-k) * b^k
where:
- n is the power to which the binomial is raised.
- k is an integer ranging from 0 to n.
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying the Theorem
Let's apply the binomial theorem to (2x - 3y)^5.
-
Identify a and b:
- a = 2x
- b = -3y
-
Determine n:
- n = 5
-
Expand the summation:
(2x - 3y)^5 = ∑(5 choose k) * (2x)^(5-k) * (-3y)^k
This expands to:
(5 choose 0) * (2x)^5 * (-3y)^0 + (5 choose 1) * (2x)^4 * (-3y)^1 + (5 choose 2) * (2x)^3 * (-3y)^2 + (5 choose 3) * (2x)^2 * (-3y)^3 + (5 choose 4) * (2x)^1 * (-3y)^4 + (5 choose 5) * (2x)^0 * (-3y)^5
-
Calculate the binomial coefficients:
- (5 choose 0) = 1
- (5 choose 1) = 5
- (5 choose 2) = 10
- (5 choose 3) = 10
- (5 choose 4) = 5
- (5 choose 5) = 1
-
Simplify each term:
- 1 * (32x^5) * 1 = 32x^5
- 5 * (16x^4) * (-3y) = -240x^4y
- 10 * (8x^3) * (9y^2) = 720x^3y^2
- 10 * (4x^2) * (-27y^3) = -1080x^2y^3
- 5 * (2x) * (81y^4) = 810xy^4
- 1 * 1 * (-243y^5) = -243y^5
-
Combine the terms:
(2x - 3y)^5 = 32x^5 - 240x^4y + 720x^3y^2 - 1080x^2y^3 + 810xy^4 - 243y^5
Final Result
Therefore, the binomial expansion of (2x - 3y)^5 is 32x^5 - 240x^4y + 720x^3y^2 - 1080x^2y^3 + 810xy^4 - 243y^5.