(2a-b)^5

Jasonbradley

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Jun 16, 2024

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Expanding (2a - b)^5 Using the Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions of the form (x + y)^n. It allows us to find the coefficients of each term in the expansion without having to multiply the entire expression out manually.

Here's how to apply the binomial theorem to expand (2a - b)^5:

The Binomial Theorem Formula

The binomial theorem states:

(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k

where:

• (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!), representing the number of ways to choose k items from a set of n items.
• ∑_(k=0)^n denotes the sum from k=0 to k=n.

Applying the Formula to (2a - b)^5

1. Identify x and y: In our expression, x = 2a and y = -b.
2. Determine n: n = 5.
3. Expand the summation: We need to calculate the terms for k = 0, 1, 2, 3, 4, and 5.

Let's calculate each term:

• k = 0: (5 choose 0) * (2a)^5 * (-b)^0 = 1 * 32a^5 * 1 = 32a^5
• k = 1: (5 choose 1) * (2a)^4 * (-b)^1 = 5 * 16a^4 * (-b) = -80a^4b
• k = 2: (5 choose 2) * (2a)^3 * (-b)^2 = 10 * 8a^3 * b^2 = 80a^3b^2
• k = 3: (5 choose 3) * (2a)^2 * (-b)^3 = 10 * 4a^2 * (-b^3) = -40a^2b^3
• k = 4: (5 choose 4) * (2a)^1 * (-b)^4 = 5 * 2a * b^4 = 10ab^4
• k = 5: (5 choose 5) * (2a)^0 * (-b)^5 = 1 * 1 * (-b^5) = -b^5

Final Expansion

Adding all the terms together, we get the expansion of (2a - b)^5:

(2a - b)^5 = 32a^5 - 80a^4b + 80a^3b^2 - 40a^2b^3 + 10ab^4 - b^5

This is the expanded form of the given expression, obtained through the application of the binomial theorem.