%5E3-expand.jpeg "(x-y)^3 Expand")
Expanding (x - y)^3
The expression (x - y)^3 represents the cube of the binomial (x - y). Expanding this expression means rewriting it as a sum of terms. There are a few ways to achieve this, but we'll focus on two common methods:
1. Using the Binomial Theorem
The Binomial Theorem provides a general formula for expanding expressions of the form (x + y)^n. For our case, n = 3, so the theorem states:
(x - y)^3 = (3 choose 0)x^3 (-y)^0 + (3 choose 1)x^2 (-y)^1 + (3 choose 2)x^1 (-y)^2 + (3 choose 3)x^0 (-y)^3
Where (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!).
Let's break down the terms:
- (3 choose 0) = 1, (3 choose 1) = 3, (3 choose 2) = 3, (3 choose 3) = 1
- (-y)^0 = 1, (-y)^1 = -y, (-y)^2 = y^2, (-y)^3 = -y^3
Substituting these values back into the equation, we get:
(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
2. Repeated Multiplication
We can also expand (x - y)^3 by multiplying the binomial by itself three times:
(x - y)^3 = (x - y)(x - y)(x - y)
First, multiply the first two binomials:
(x - y)(x - y) = x^2 - xy - xy + y^2 = x^2 - 2xy + y^2
Now, multiply the result by the third binomial:
(x^2 - 2xy + y^2)(x - y) = x^3 - 2x^2y + xy^2 - x^2y + 2xy^2 - y^3 = x^3 - 3x^2y + 3xy^2 - y^3
Conclusion
Both methods lead to the same expanded form: (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3. This expression demonstrates the pattern that emerges when expanding a binomial cubed. It is important to remember the signs and coefficients, which are determined by the binomial theorem or by careful multiplication.