2 Find the quotient and remainder when \(2 x ^ { 4 } - 27\) is divided by \(x ^ { 2 } + x + 3\).
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Question 2:
Answer Marks
Guidance
Answer Mark
Guidance
Divide to obtain quotient \(2x^2 \pm 2x + k\) \((k \neq 0)\) M1
Obtain result in answer column, together with a linear polynomial or a constant as remainder
Obtain [quotient] \(2x^2 - 2x - 4\) A1
Allow unless quotient and remainder interchanged, then A0 A1
Obtain [remainder] \(10x - 15\) A1
Allow \((x^2 + x + 3)(2x^2 - 2x - 4) + 10x - 15\)
Alternative Method:
Answer Marks
Guidance
Answer Mark
Guidance
Expand \((x^2+x+3)(Ax^2+Bx+C)+(Dx+E)\) and reach \(A=2\), \(B=\pm2\), \(C=k\) M1
Solve all 3 equations for \(A\), \(B\) and \(C\); if correct \(A=2\), \(A+B=0\), \(3A+B+C=0\), \(3B+C+D=0\), \(3C+E=-27\)
Obtain [quotient] \(2x^2 - 2x - 4\) A1
Allow unless quotient and remainder interchanged, then A0 A1
Obtain [remainder] \(10x - 15\) A1
Allow \((x^2+x+3)(2x^2-2x-4)+10x-15\)
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## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| Divide to obtain quotient $2x^2 \pm 2x + k$ $(k \neq 0)$ | M1 | Obtain result in answer column, together with a linear polynomial or a constant as remainder |
| Obtain [quotient] $2x^2 - 2x - 4$ | A1 | Allow unless quotient and remainder interchanged, then A0 A1 |
| Obtain [remainder] $10x - 15$ | A1 | Allow $(x^2 + x + 3)(2x^2 - 2x - 4) + 10x - 15$ |
**Alternative Method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Expand $(x^2+x+3)(Ax^2+Bx+C)+(Dx+E)$ and reach $A=2$, $B=\pm2$, $C=k$ | M1 | Solve all 3 equations for $A$, $B$ and $C$; if correct $A=2$, $A+B=0$, $3A+B+C=0$, $3B+C+D=0$, $3C+E=-27$ |
| Obtain [quotient] $2x^2 - 2x - 4$ | A1 | Allow unless quotient and remainder interchanged, then A0 A1 |
| Obtain [remainder] $10x - 15$ | A1 | Allow $(x^2+x+3)(2x^2-2x-4)+10x-15$ |
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2 Find the quotient and remainder when $2 x ^ { 4 } - 27$ is divided by $x ^ { 2 } + x + 3$.\\
\hfill \mbox{\textit{CAIE P3 2023 Q2 [3]}}