Question

Expand (2x-3y)^(5) using Binomial theorem.

Answer 1

To expand (2x - 3y)^5, we can use the binomial theorem. Applying the binomial theorem formula, we get the expanded form: 32x^5 - 240x^4y + 720x^3y^2 - 1080x^2y^3 + 810xy^4 + 243y^5.

Step-by-step explanation:

The binomial theorem provides a way to expand expressions of the form (a + b)^n, where a and b are constants and n is a positive integer. To expand (2x - 3y)^5, we can use the binomial theorem formula:

(a + b)^n = C(n, 0)a^n + C(n, 1)a^(n-1)b + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)ab^(n-1) + C(n, n)b^n

Applying this formula to (2x - 3y)^5:

(2x - 3y)^5 = C(5, 0)(2x)^5 + C(5, 1)(2x)^4(-3y) + C(5, 2)(2x)^3(-3y)^2 + C(5, 3)(2x)^2(-3y)^3 + C(5, 4)(2x)(-3y)^4 + C(5, 5)(-3y)^5

Simplifying each term, we get:

1. C(5, 0)(2x)^5 = 1(32x^5) = 32x^5
2. C(5, 1)(2x)^4(-3y) = 5(16x^4)(-3y) = -240x^4y
3. C(5, 2)(2x)^3(-3y)^2 = 10(8x^3)(9y^2) = 720x^3y^2
4. C(5, 3)(2x)^2(-3y)^3 = 10(4x^2)(-27y^3) = -1080x^2y^3
5. C(5, 4)(2x)(-3y)^4 = 5(2x)(81y^4) = 810xy^4
6. C(5, 5)(-3y)^5 = 1(243y^5) = 243y^5

Adding all the terms together, we get the expanded form of (2x - 3y)^5:

32x^5 - 240x^4y + 720x^3y^2 - 1080x^2y^3 + 810xy^4 + 243y^5