rewrite the following expression in expanded form and simplify completely.

\(3x - y)^5\

Question

rewrite the following expression in expanded form and simplify completely.

\(3x - y)^5\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Binomial Theorem", "Binomial Expansion", "Binomial Coefficient" ], "new_concepts": [], "current_concepts": [ "Binomial Theorem", "Binomial Expansion", "Binomial Coefficient" ] } ## 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 $$ \begin{aligned} \text{Term 1: } & 1 \cdot 243x^5 \cdot 1 = 243x^5 \ \text{Term 2: } & 5 \cdot 81x^4 \cdot (-y) = -405x^4y \ \text{Term 3: } & 10 \cdot 27x^3 \cdot y^2 = 270x^3y^2 \ \text{Term 4: } & 10 \cdot 9x^2 \cdot (-y^3) = -90x^2y^3 \ \text{Term 5: } & 5 \cdot 3x \cdot y^4 = 15xy^4 \ \text{Term 6: } & 1 \cdot 1 \cdot (-y^5) = -y^5 \end{aligned} $$ $$243x^5 - 405x^4y + 270x^3y^2 - 90x^2y^3 + 15xy^4 - y^5$$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Binomial Expansion" ] }

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QUESTION IMAGE

Question

Explanation:

Apply the Binomial Theorem formula

Expand the terms using binomial coefficients

Simplify each term individually

Answer: