| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Topic | Factor & Remainder Theorem |
| Type | Fully specified polynomial: verify factor and solve |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring only routine procedures: substitute x=1 to verify it's a root, perform polynomial division by (x-1), factorise the resulting quadratic, then state the solutions. All steps are standard textbook exercises with no problem-solving or insight required, making it easier than average but not trivial since it involves multiple algebraic manipulations. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
**Part (i)**
- $f(1) = 0$ clearly shown: B1
- Attempt method for division by $(x - 1)$ only: M1
- Obtain $x^2 - 2x - 15$: A1
- Obtain $(x-1)(x+3)(x-5)$: A1 **[4]**
**Part (ii)**
- State any two correct roots: B1$\checkmark$
- State $x = -3, 1, 5$: B1 **[2]**
**[Total: 6]**2 Let $\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15$.\\
(i) Show that $\mathrm { f } ( 1 ) = 0$ and hence factorise $x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15$ completely.\\
(ii) Hence solve the equation $x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15 = 0$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q2 [6]}}