%5E3-expand.jpeg "(1-y)^3 Expand")
Expanding (1-y)^3
In algebra, expanding an expression means to multiply it out and simplify it as much as possible. In this article, we will expand the expression (1-y)^3.
The Binomial Theorem
To expand (1-y)^3, we can use the Binomial Theorem, which states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, a and b are variables, and $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
Expanding (1-y)^3
Using the Binomial Theorem, we can expand (1-y)^3 as follows:
$(1-y)^3 = \sum_{k=0}^3 \binom{3}{k} 1^{3-k} (-y)^k$
Simplifying the expression, we get:
$(1-y)^3 = \binom{3}{0} 1^3 (-y)^0 + \binom{3}{1} 1^2 (-y)^1 + \binom{3}{2} 1^1 (-y)^2 + \binom{3}{3} 1^0 (-y)^3$
Simplifying further, we get:
$(1-y)^3 = 1 - 3y + 3y^2 - y^3$
Final Answer
Therefore, the expanded form of (1-y)^3 is:
$(1-y)^3 = \boxed{1 - 3y + 3y^2 - y^3}$
We can verify this by plugging in some values of y and checking that the equation holds true.
