what expression is equivalent to $(5z^2 + 3z + 2)^2$?

* $5z^4 + 3z^2 + 4$
* $5z^4 + 9z^2 + 4$
* $25z^4 + 30z^3 + 19z^2 + 12z + 4$
* $25z^4 + 30z^3 + 29z^2 + 12z + 4$

Question

what expression is equivalent to $(5z^2 + 3z + 2)^2$?

* $5z^4 + 3z^2 + 4$
* $5z^4 + 9z^2 + 4$
* $25z^4 + 30z^3 + 19z^2 + 12z + 4$
* $25z^4 + 30z^3 + 29z^2 + 12z + 4$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Multiplication" ], "new_concepts": [ "Trinomial Squaring" ], "current_concepts": [ "Polynomial Multiplication", "Trinomial Squaring" ] } ## Set up the polynomial multiplication $$ (5z^2 + 3z + 2)^2 = (5z^2 + 3z + 2)(5z^2 + 3z + 2) $$ ## Distribute and expand the terms $$ \begin{aligned} (5z^2 + 3z + 2)(5z^2 + 3z + 2) = & \ 5z^2(5z^2 + 3z + 2) \ & + 3z(5z^2 + 3z + 2) \ & + 2(5z^2 + 3z + 2) \end{aligned} $$ $$ \begin{aligned} = & \ (25z^4 + 15z^3 + 10z^2) \ & + (15z^3 + 9z^2 + 6z) \ & + (10z^2 + 6z + 4) \end{aligned} $$ ## Combine like terms $$ \begin{aligned} & 25z^4 + (15 + 15)z^3 + (10 + 9 + 10)z^2 + (6 + 6)z + 4 \ = & \ 25z^4 + 30z^3 + 29z^2 + 12z + 4 \end{aligned} $$

(A) $5z^4 + 3z^2 + 4$

(B) $5z^4 + 9z^2 + 4$

(C) $25z^4 + 30z^3 + 19z^2 + 12z + 4$

(D) $25z^4 + 30z^3 + 29z^2 + 12z + 4$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Multiplication" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the polynomial multiplication

Distribute and expand the terms

Combine like terms

Answer: