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 algebraic manipulation across multiple steps, it's a standard textbook exercise with no conceptual challenges—students simply execute the mechanical process of dividing by a quadratic divisor and expressing the result in quotient-remainder form.
1 The polynomial \(\mathrm { f } ( x )\) is defined by
$$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - 6 x + 11$$
Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(\left( x ^ { 2 } + 2 \right)\).
Divide at least as far as the \(x\) term in the quotient
M1
Allow use of \((x^2+2)(x^2+ax+b)+cx+d\)
Obtain at least \(x^2-3x\)
A1
Obtain \(x^2-3x+3\) and remainder 5
A1
Total
3
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Divide at least as far as the $x$ term in the quotient | M1 | Allow use of $(x^2+2)(x^2+ax+b)+cx+d$ |
| Obtain at least $x^2-3x$ | A1 | |
| Obtain $x^2-3x+3$ and remainder 5 | A1 | |
| **Total** | **3** | |
---
1 The polynomial $\mathrm { f } ( x )$ is defined by
$$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - 6 x + 11$$
Find the quotient and remainder when $\mathrm { f } ( x )$ is divided by $\left( x ^ { 2 } + 2 \right)$.\\
\hfill \mbox{\textit{CAIE P2 2019 Q1 [3]}}