(x-2y)^7

4 min read Jun 17, 2024
(x-2y)^7

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

The Binomial Theorem provides a systematic way to expand expressions of the form (x + y)^n. In this case, we'll be expanding (x - 2y)^7.

The Binomial Theorem Formula

The Binomial Theorem states that:

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

Where:

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

Expanding (x - 2y)^7

Let's apply the Binomial Theorem to our expression:

  1. Identify n: In this case, n = 7.

  2. Expand the sum: We'll need to calculate the terms for k = 0, 1, 2, 3, 4, 5, 6, and 7.

  3. Calculate the binomial coefficients: For each value of k, we'll calculate (7 choose k).

  4. Substitute x and y: In our expression, x = x and y = -2y.

Let's break down the expansion term by term:

k = 0:

  • (7 choose 0) = 1
  • x^(7-0) = x^7
  • (-2y)^0 = 1
  • Term: x^7

k = 1:

  • (7 choose 1) = 7
  • x^(7-1) = x^6
  • (-2y)^1 = -2y
  • Term: -14x^6y

k = 2:

  • (7 choose 2) = 21
  • x^(7-2) = x^5
  • (-2y)^2 = 4y^2
  • Term: 84x^5y^2

k = 3:

  • (7 choose 3) = 35
  • x^(7-3) = x^4
  • (-2y)^3 = -8y^3
  • Term: -280x^4y^3

k = 4:

  • (7 choose 4) = 35
  • x^(7-4) = x^3
  • (-2y)^4 = 16y^4
  • Term: 560x^3y^4

k = 5:

  • (7 choose 5) = 21
  • x^(7-5) = x^2
  • (-2y)^5 = -32y^5
  • Term: -672x^2y^5

k = 6:

  • (7 choose 6) = 7
  • x^(7-6) = x
  • (-2y)^6 = 64y^6
  • Term: 448xy^6

k = 7:

  • (7 choose 7) = 1
  • x^(7-7) = 1
  • (-2y)^7 = -128y^7
  • Term: -128y^7

Final Expanded Form

Combining all the terms, the complete expansion of (x - 2y)^7 is:

(x - 2y)^7 = x^7 - 14x^6y + 84x^5y^2 - 280x^4y^3 + 560x^3y^4 - 672x^2y^5 + 448xy^6 - 128y^7

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