Moderate -0.3 This is a polynomial identity question requiring students to expand both sides and equate coefficients. While it involves algebraic manipulation across multiple terms, it's a straightforward mechanical process with no conceptual difficulty—students simply expand, collect like terms, and solve the resulting system of equations. Slightly easier than average due to its routine nature.
2 Given that
$$( x - p ) \left( 2 x ^ { 2 } + 9 x + 10 \right) = \left( x ^ { 2 } - 4 \right) ( 2 x + q )$$
for all values of \(x\), find the constants \(p\) and \(q\).
Attempt to expand both sides OR substitute 2 values of \(x\) into both expressions OR express at least one side as product of three factors; If expanding, minimum of 5 terms on LHS and 3 terms on RHS
DM1
Valid method to obtain either \(p\) or \(q\); If comparing coefficients, must be of corresponding terms
\(p = 2\) and \(q = 5\)
A1
3 marks (total 3); Both values correct; SR Spotted solutions B1 one correct B2 other correct
# Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| $2x^3 + 9x^2 - 2px^2 - 9px + 10x - 10p = 2x^3 + qx^2 - 8x - 4q$ | M1* | Attempt to expand both sides **OR** substitute 2 values of $x$ into both expressions **OR** express at least one side as product of three factors; If expanding, minimum of 5 terms on LHS and 3 terms on RHS |
| | DM1 | Valid method to obtain either $p$ or $q$; If comparing coefficients, must be of corresponding terms |
| $p = 2$ and $q = 5$ | A1 | **3 marks** (total **3**); Both values correct; **SR** Spotted solutions **B1** one correct **B2** other correct |
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2 Given that
$$( x - p ) \left( 2 x ^ { 2 } + 9 x + 10 \right) = \left( x ^ { 2 } - 4 \right) ( 2 x + q )$$
for all values of $x$, find the constants $p$ and $q$.
\hfill \mbox{\textit{OCR C1 2011 Q2 [3]}}