%5E3.jpeg "(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