(2x-3y)^5 Binomial Expansion

Jasonbradley

3 min read

·

Jun 16, 2024

28

Share

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.

1. Identify a and b:

   • a = 2x
   • b = -3y

2. Determine n:

   • n = 5

3. 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

4. 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

5. 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

6. 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.