Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 \)
Solution
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:
\[
x^3 - 3x^2y + 3xy^2 - y^3
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To expand \( (x-y)^{3} \), we use the binomial theorem which states that \( (a-b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^{k} \). For \( n=3 \), \( a=x \), and \( b=y \): \[ (x-y)^{3} = x^{3} - 3x^{2}y + 3xy^{2} - y^{3} \] So the expansion of \( (x-y)^{3} \) is \( x^{3} - 3x^{2}y + 3xy^{2} - y^{3} \). In simplified form, it includes a cubic term of \( x \), a quadratic and linear mix of \( x \) and \( y \), and a negative cubic term of \( y \).
