| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.8 This is a straightforward AS-level question testing standard techniques: part (a) requires simple substitution into the factor theorem (f(3)=0), and part (b) involves routine polynomial division followed by checking the discriminant of a quadratic. Both parts are textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) | States or uses f((cid:14)3)(cid:32)0 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| factor | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | Begins division or factorisation so x | |
| 4x3 – 12x2 + 2x – 6 = (x (cid:237)(cid:22))((cid:23)x2 (cid:14)(cid:171)) | M1 | 2.1 |
| 4x3 – 12x2 + 2x – 6 = (x (cid:237)(cid:22))((cid:23)x2 + 2) | A 1 | 1 . 1b |
| Answer | Marks | Guidance |
|---|---|---|
| square or discriminant | M1 | 2.1 |
| Answer | Marks | Guidance |
|---|---|---|
| So x = 3 is the only real root of f(x) = 0 * | A1* | 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Scheme | Marks |
Question 4:
--- 4(a) ---
4(a) | States or uses f((cid:14)3)(cid:32)0 | M1 | 1.1b
4(3)3 – 12(3)2 + 2(3) – 6 = 108 – 108 + 6 – 6 = 0 and so (x – 3) is a
factor | A1 | 1.1b
(2)
(b) | Begins division or factorisation so x
4x3 – 12x2 + 2x – 6 = (x (cid:237)(cid:22))((cid:23)x2 (cid:14)(cid:171)) | M1 | 2.1
4x3 – 12x2 + 2x – 6 = (x (cid:237)(cid:22))((cid:23)x2 + 2) | A 1 | 1 . 1b
Considers the roots of their quadratic function using completion of
square or discriminant | M1 | 2.1
(4x2 + 2) = 0 has no real roots with a reason (e.g. negative number
does not have a real square root, or 4x2 (cid:14)2(cid:33)0for all x
So x = 3 is the only real root of f(x) = 0 * | A1* | 2.4
(4)
(6 marks)
Notes:
(a)
M1: States or uses f (+3) = 0
A1: See correct work evaluating and achieving zero, together with correct conclusion
(b)
M1: Needs to have (x – 3) and first term of quadratic correct
A1: Must be correct – may further factorise to 2(x (cid:237) 3)(2x2 + 1)
M1: Considers their quadratic for no real roots by use of completion of the square or
consideration of discriminant then
A1*: A correct explanation
Question | Scheme | Marks | AOs$$f(x) = 4x^3 - 12x^2 + 2x - 6$$
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $(x - 3)$ is a factor of $f(x)$.
[2]
\item Hence show that $3$ is the only real root of the equation $f(x) = 0$
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 Q4 [6]}}