Simplify and expand 1) \( (x-y)^{3} \)

To expand \( (x - y)^3 \), you can use the binomial theorem or perform the expansion step by step. Here's the expanded form: \[ (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 \] **Step-by-Step Expansion:** 1. **Start with the binomial:** \[ (x - y)^3 = (x - y)(x - y)(x - y) \] 2. **First, multiply two of the binomials:** \[ (x - y)(x - y) = x^2 - 2xy + y^2 \] 3. **Now, multiply the result by the remaining binomial:** \[ (x^2 - 2xy + y^2)(x - y) \] 4. **Distribute each term in the first polynomial by each term in the second:** \[ x^2 \cdot x = x^3 \] \[ x^2 \cdot (-y) = -x^2y \] \[ -2xy \cdot x = -2x^2y \] \[ -2xy \cdot (-y) = 2xy^2 \] \[ y^2 \cdot x = xy^2 \] \[ y^2 \cdot (-y) = -y^3 \] 5. **Combine like terms:** \[ x^3 - x^2y - 2x^2y + 2xy^2 + xy^2 - y^3 = x^3 - 3x^2y + 3xy^2 - y^3 \] So, the simplified and expanded form of \( (x - y)^3 \) is: \[ x^3 - 3x^2y + 3xy^2 - y^3 \]