Certainly! Let's expand the expression [tex]\((x - y)^3\)[/tex].
1. Start with the given expression:
[tex]\[
(x - y)^3
\][/tex]
2. Use the binomial theorem to expand the expression. The binomial theorem states:
[tex]\[
(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k
\][/tex]
For our particular case, [tex]\(a = x\)[/tex], [tex]\(b = y\)[/tex], and [tex]\(n = 3\)[/tex]. So we have:
[tex]\[
(x - y)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} (-y)^k
\][/tex]
3. Apply the binomial coefficients:
[tex]\[
\begin{aligned}
(x - y)^3 &= \binom{3}{0} x^{3} (-y)^{0} + \binom{3}{1} x^{2} (-y)^{1} + \binom{3}{2} x^{1} (-y)^{2} + \binom{3}{3} x^{0} (-y)^{3} \\
&= 1 \cdot x^{3} \cdot 1 - 3 \cdot x^{2} \cdot y + 3 \cdot x \cdot y^{2} - 1 \cdot y^{3} \\
&= x^3 - 3x^2y + 3xy^2 - y^3
\end{aligned}
\][/tex]
4. Combine the terms:
[tex]\[
(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
\][/tex]
So, the expanded form of [tex]\((x-y)^3\)[/tex] is:
[tex]\[
x^3 - 3x^2y + 3xy^2 - y^3
\][/tex]
By carefully expanding the binomial expression, we arrive at the final result.
