Question

rewrite the following expression in expanded form and simplify completely. $(x - y)^4$

Explanation:

Step1: Use binomial theorem

The binomial theorem states $(a-b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k$. For $(x-y)^4$, $a=x$, $b=y$, $n=4$.
$$(x-y)^4 = \binom{4}{0}x^4(-y)^0 + \binom{4}{1}x^3(-y)^1 + \binom{4}{2}x^2(-y)^2 + \binom{4}{3}x^1(-y)^3 + \binom{4}{4}x^0(-y)^4$$

Step2: Calculate binomial coefficients

$\binom{4}{0}=1$, $\binom{4}{1}=4$, $\binom{4}{2}=6$, $\binom{4}{3}=4$, $\binom{4}{4}=1$. Substitute these values:
$$=1 \cdot x^4 \cdot 1 + 4 \cdot x^3 \cdot (-y) + 6 \cdot x^2 \cdot y^2 + 4 \cdot x \cdot (-y^3) + 1 \cdot 1 \cdot y^4$$

Step3: Simplify each term

$$=x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4$$

Answer:

$x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4$