(1-2n)^3

2 min read Jun 16, 2024
(1-2n)^3

Expanding (1 - 2n)^3

The expression (1 - 2n)^3 represents the cube of the binomial (1 - 2n). To expand this expression, we can use the Binomial Theorem or simply multiply it out.

Using the Binomial Theorem

The Binomial Theorem states that for any real numbers x and y, and any non-negative integer n:

(x + y)^n = x^n + nx^(n-1)y + (n(n-1)/2!)x^(n-2)y^2 + ... + nyx^(n-1) + y^n

Applying this to our expression (1 - 2n)^3, we have:

  • x = 1
  • y = -2n
  • n = 3

Substituting these values into the Binomial Theorem, we get:

(1 - 2n)^3 = 1^3 + 3(1^2)(-2n) + 3(1)(-2n)^2 + (-2n)^3

Simplifying this expression, we obtain:

(1 - 2n)^3 = 1 - 6n + 12n^2 - 8n^3

Expanding Directly

We can also expand (1 - 2n)^3 by multiplying it out directly:

(1 - 2n)^3 = (1 - 2n)(1 - 2n)(1 - 2n)

First, we multiply the first two factors:

(1 - 2n)(1 - 2n) = 1 - 4n + 4n^2

Then, we multiply this result by the remaining factor (1 - 2n):

(1 - 4n + 4n^2)(1 - 2n) = 1 - 6n + 12n^2 - 8n^3

This gives us the same result as we obtained using the Binomial Theorem.

Summary

Therefore, the expanded form of (1 - 2n)^3 is:

1 - 6n + 12n^2 - 8n^3

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