%5E3-expand.jpeg "(x+y)^3 Expand")
Expanding (x+y)^3
In algebra, expanding an expression means to simplify it by removing any parentheses or grouping symbols. In this article, we will learn how to expand the expression (x+y)^3.
What is the formula?
The formula to expand (x+y)^n is given by the Binomial Theorem, which states:
$(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$
where n is a positive integer, and \binom{n}{k} is the binomial coefficient, which can be calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
In our case, n=3, so we can plug this value into the formula to get:
$(x+y)^3 = \sum_{k=0}^3 \binom{3}{k} x^{3-k} y^k$
Expanding the expression
Now, let's expand the expression (x+y)^3 by plugging in the values of k from 0 to 3:
k=0
$\binom{3}{0} x^{3-0} y^0 = \frac{3!}{0!(3-0)!} x^3 y^0 = 1 \cdot x^3 \cdot 1 = x^3$
k=1
$\binom{3}{1} x^{3-1} y^1 = \frac{3!}{1!(3-1)!} x^2 y^1 = 3 \cdot x^2 \cdot y = 3x^2y$
k=2
$\binom{3}{2} x^{3-2} y^2 = \frac{3!}{2!(3-2)!} x^1 y^2 = 3 \cdot x^1 \cdot y^2 = 3xy^2$
k=3
$\binom{3}{3} x^{3-3} y^3 = \frac{3!}{3!(3-3)!} x^0 y^3 = 1 \cdot x^0 \cdot y^3 = y^3$
Final result
Adding up all the terms, we get:
$(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
And that's the expanded form of the expression (x+y)^3!
