Question

rewrite the following expression in expanded form and simplify completely.

\\(3x - y)^5\\

Explanation:

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<reasoning>

Apply the Binomial Theorem formula

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
\[
a = 3x, \quad b = -y, \quad n = 5
\]

Expand the terms using binomial coefficients

\[

$$\begin{aligned} (3x - y)^5 = & \binom{5}{0} (3x)^5 (-y)^0 + \binom{5}{1} (3x)^4 (-y)^1 + \binom{5}{2} (3x)^3 (-y)^2 \\ & + \binom{5}{3} (3x)^2 (-y)^3 + \binom{5}{4} (3x)^1 (-y)^4 + \binom{5}{5} (3x)^0 (-y)^5 \end{aligned}$$

\]

Simplify each term individually

\[

$$\begin{aligned} \text{Term 1: } & 1 \cdot 243x^5 \cdot 1 = 243x^5 \\ \text{Term 2: } & 5 \cdot 81x^4 \cdot (-y) = -405x^4y \\ \text{Term 3: } & 10 \cdot 27x^3 \cdot y^2 = 270x^3y^2 \\ \text{Term 4: } & 10 \cdot 9x^2 \cdot (-y^3) = -90x^2y^3 \\ \text{Term 5: } & 5 \cdot 3x \cdot y^4 = 15xy^4 \\ \text{Term 6: } & 1 \cdot 1 \cdot (-y^5) = -y^5 \end{aligned}$$

\]
</reasoning>

<answer>
\[243x^5 - 405x^4y + 270x^3y^2 - 90x^2y^3 + 15xy^4 - y^5\]
</answer>

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Answer:

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<reasoning>

Apply the Binomial Theorem formula

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
\[
a = 3x, \quad b = -y, \quad n = 5
\]

Expand the terms using binomial coefficients

\[

$$\begin{aligned} (3x - y)^5 = & \binom{5}{0} (3x)^5 (-y)^0 + \binom{5}{1} (3x)^4 (-y)^1 + \binom{5}{2} (3x)^3 (-y)^2 \\ & + \binom{5}{3} (3x)^2 (-y)^3 + \binom{5}{4} (3x)^1 (-y)^4 + \binom{5}{5} (3x)^0 (-y)^5 \end{aligned}$$

\]

Simplify each term individually

\[

$$\begin{aligned} \text{Term 1: } & 1 \cdot 243x^5 \cdot 1 = 243x^5 \\ \text{Term 2: } & 5 \cdot 81x^4 \cdot (-y) = -405x^4y \\ \text{Term 3: } & 10 \cdot 27x^3 \cdot y^2 = 270x^3y^2 \\ \text{Term 4: } & 10 \cdot 9x^2 \cdot (-y^3) = -90x^2y^3 \\ \text{Term 5: } & 5 \cdot 3x \cdot y^4 = 15xy^4 \\ \text{Term 6: } & 1 \cdot 1 \cdot (-y^5) = -y^5 \end{aligned}$$

\]
</reasoning>

<answer>
\[243x^5 - 405x^4y + 270x^3y^2 - 90x^2y^3 + 15xy^4 - y^5\]
</answer>

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