| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Remainder condition then further work |
| Difficulty | Moderate -0.3 This is a straightforward C2 question testing standard Factor/Remainder Theorem applications. Part (a) uses f(-3)=0 to find p (simple substitution), part (b) verifies f(2)=50 (routine calculation), and part (c) requires factorizing and solving a quadratic using the formula. All steps are mechanical with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| \(-27 + 63 - 3p - 6 = 0, \quad p = 10\) | M1 A1 |
| Answer | Marks |
|---|---|
| remainder \(= f(2) = 8 + 28 + 20 - 6 = 50\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(x = -3\) is a solution \(\therefore (x + 3)\) is a factor | B1 |
| \(x + 3 \mid x^3 + 7x^2 + 10x - 6\) with quotient \(x^2 + 4x - 2\) | M1 A1 |
| \((x + 3)(x^2 + 4x - 2) = 0\) | |
| \(x = -3\) or \(x^2 + 4x - 2 = 0\) | |
| other solutions: \(x = \frac{-4 \pm \sqrt{16+8}}{2} = -4.45, 0.45\) | M1 A1 |
**Part (a)**
$-27 + 63 - 3p - 6 = 0, \quad p = 10$ | M1 A1 |
**Part (b)**
remainder $= f(2) = 8 + 28 + 20 - 6 = 50$ | M1 A1 |
**Part (c)**
$x = -3$ is a solution $\therefore (x + 3)$ is a factor | B1 |
$x + 3 \mid x^3 + 7x^2 + 10x - 6$ with quotient $x^2 + 4x - 2$ | M1 A1 |
$(x + 3)(x^2 + 4x - 2) = 0$ |
$x = -3$ or $x^2 + 4x - 2 = 0$ |
other solutions: $x = \frac{-4 \pm \sqrt{16+8}}{2} = -4.45, 0.45$ | M1 A1 |
---5. Given that
$$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6 ,$$
and that $x = - 3$ is a solution to the equation $\mathrm { f } ( x ) = 0$,
\begin{enumerate}[label=(\alph*)]
\item find the value of the constant $p$,
\item show that when $\mathrm { f } ( x )$ is divided by $( x - 2 )$ there is a remainder of 50 ,
\item find the other solutions to the equation $\mathrm { f } ( x ) = 0$, giving your answers to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q5 [9]}}