\p = 2b^2 - 1\
\q = 8b + b^2 - 6\
complete the expression in terms of $p$ and $q$ so the result is $-5b^2 + 8b - 3$.
drag the terms to the correct location on the expression. not all terms will be used.
options: $p$, $q$, $2p$, $2q$, $3p$

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Polynomial Operations", "System of Polynomial Equations" ], "current_concepts": [ "Polynomial Operations", "System of Polynomial Equations" ] } ## Express the target polynomial in terms of P and Q $$ \begin{aligned} P &= 2b^2 - 1 \ Q &= 8b + b^2 - 6 = b^2 + 8b - 6 \ T &= -5b^2 + 8b - 3 \end{aligned} $$ ## Set up a linear combination of P and Q $$ \begin{aligned} xP + yQ &= T \ x(2b^2 - 1) + y(b^2 + 8b - 6) &= -5b^2 + 8b - 3 \ (2x + y)b^2 + 8yb - (x + 6y) &= -5b^2 + 8b - 3 \end{aligned} $$ ## Solve for the coefficients x and y $$ \begin{aligned} 8y &= 8 \implies y = 1 \ 2x + y &= -5 \implies 2x + 1 = -5 \implies 2x = -6 \implies x = -3 \ -(x + 6y) &= -(-3 + 6(1)) = -3 \quad (\text{consistent}) \ T &= -3P + Q = Q - 3P \end{aligned} $$ Complete the expression in terms of $P$ and $Q$ so the result is $-5b^2 + 8b - 3$. $Q$ $-$ $3P$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Operations" ] }

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Question

Explanation:

Express the target polynomial in terms of P and Q

Set up a linear combination of P and Q

Solve for the coefficients x and y

Answer: