Moderate -0.5 This is a straightforward polynomial long division problem requiring systematic application of the division algorithm. While it involves a quartic polynomial and requires careful arithmetic across multiple steps, it's a standard textbook exercise with no conceptual difficulty or problem-solving insight needed—just methodical execution of a learned procedure.
Commence division and reach partial quotient of the form \(x^2 \pm 3x\), or \(x^4 - 3x^3 + 9x^2 - 12x + 27 = (x^2+5)(Ax^2+Bx+C)+Dx+E\), or \(Ax^4 + Bx^3 + (5A+C)x^2 + 5Bx + 5C\), and reach \(A=1\) and \(B = \pm 3\)
M1
Obtain quotient \(x^2 - 3x + 4\)
A1
\(A=1\), \(B=-3\); \([5A+C=9\) so \(C=4\); \(5B+D=-12\) so \(D=3\); \(5C+E=27\) so \(E=7]\). A pair of incorrect statements 'remainder \(x^2-3x+4\)' and 'quotient \(3x+7\)' score M1 A1 A0.
Obtain remainder \(3x + 7\)
A1
Total: 3 marks
3
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Commence division and reach partial quotient of the form $x^2 \pm 3x$, or $x^4 - 3x^3 + 9x^2 - 12x + 27 = (x^2+5)(Ax^2+Bx+C)+Dx+E$, or $Ax^4 + Bx^3 + (5A+C)x^2 + 5Bx + 5C$, and reach $A=1$ and $B = \pm 3$ | M1 | |
| Obtain quotient $x^2 - 3x + 4$ | A1 | $A=1$, $B=-3$; $[5A+C=9$ so $C=4$; $5B+D=-12$ so $D=3$; $5C+E=27$ so $E=7]$. A pair of incorrect statements 'remainder $x^2-3x+4$' and 'quotient $3x+7$' score **M1 A1 A0**. |
| Obtain remainder $3x + 7$ | A1 | |
| **Total: 3 marks** | **3** | |