(2x-3)^7

Jasonbradley

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Jun 16, 2024

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Expanding (2x - 3)^7 using the Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. Let's apply it to expand (2x - 3)^7.

The Binomial Theorem

The Binomial Theorem states:

(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k

Where:

• n is a positive integer representing the power.
• k is an integer ranging from 0 to n.
• (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).

Expanding (2x - 3)^7

Let's break down the expansion of (2x - 3)^7 using the Binomial Theorem:

1. Identify a and b: In this case, a = 2x and b = -3.

2. Determine n: The power is 7, so n = 7.

3. Calculate binomial coefficients: We need to calculate (7 choose k) for k = 0, 1, 2, ... , 7. Here are the values:

   • (7 choose 0) = 1
   • (7 choose 1) = 7
   • (7 choose 2) = 21
   • (7 choose 3) = 35
   • (7 choose 4) = 35
   • (7 choose 5) = 21
   • (7 choose 6) = 7
   • (7 choose 7) = 1

4. Apply the Binomial Theorem:

   (2x - 3)^7 = (7 choose 0) * (2x)^7 * (-3)^0 + (7 choose 1) * (2x)^6 * (-3)^1 + (7 choose 2) * (2x)^5 * (-3)^2 + (7 choose 3) * (2x)^4 * (-3)^3 + (7 choose 4) * (2x)^3 * (-3)^4 + (7 choose 5) * (2x)^2 * (-3)^5 + (7 choose 6) * (2x)^1 * (-3)^6 + (7 choose 7) * (2x)^0 * (-3)^7

5. Simplify:

   (2x - 3)^7 = 128x^7 - 1344x^6 + 5040x^5 - 10080x^4 + 10080x^3 - 5670x^2 + 1701x - 2187

Therefore, the expanded form of (2x - 3)^7 is 128x^7 - 1344x^6 + 5040x^5 - 10080x^4 + 10080x^3 - 5670x^2 + 1701x - 2187.