select the correct answer.

consider the polynomials given below.
\
\begin{aligned}
p &= x^4 + 3x^3 + 2x^2 - x + 2 \\
q &= (x^3 + 2x^2 + 3)(x^2 - 2)
\end{aligned}
\

determine the operation that results in the simplified expression below.
\
x^5 + x^4 - 5x^3 - 3x^2 + x - 8
\

- $pq$
- $q - p$
- $p + q$
- $p - q$

Question

select the correct answer.

consider the polynomials given below.
\
\begin{aligned}
p &= x^4 + 3x^3 + 2x^2 - x + 2 \\
q &= (x^3 + 2x^2 + 3)(x^2 - 2)
\end{aligned}
\

determine the operation that results in the simplified expression below.
\
x^5 + x^4 - 5x^3 - 3x^2 + x - 8
\

- $pq$
- $q - p$
- $p + q$
- $p - q$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Addition", "Polynomial Factoring", "Combining Like Terms" ], "new_concepts": [], "current_concepts": [ "Polynomial Addition", "Polynomial Division", "Combining Like Terms" ] } ## Expand the polynomial Q $$ \begin{aligned} Q &= (x^3 + 2x^2 + 3)(x^2 - 2) \ &= x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2) \ &= x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6 \ &= x^5 + 2x^4 - 2x^3 - x^2 - 6 \end{aligned} $$ ## Test the operation Q - P $$ \begin{aligned} Q - P &= (x^5 + 2x^4 - 2x^3 - x^2 - 6) - (x^4 + 3x^3 + 2x^2 - x + 2) \ &= x^5 + (2x^4 - x^4) + (-2x^3 - 3x^3) + (-x^2 - 2x^2) - (-x) + (-6 - 2) \ &= x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \end{aligned} $$ ## Verify the target expression $$ x^5 + x^4 - 5x^3 - 3x^2 + x - 8 = Q - P $$

(A) $PQ$

(B) $Q - P$

(C) $P + Q$

(D) $P - Q$

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

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QUESTION IMAGE

Question

Explanation:

Expand the polynomial Q

Test the operation Q - P

Verify the target expression

Answer: