Easy -1.2 This is a straightforward application of the factor theorem requiring only substitution of x=1 into f(x), setting equal to zero, and solving a simple linear equation for a. It's a routine single-step problem testing basic recall of the factor theorem with minimal algebraic manipulation.
1.
$$f ( x ) = a x ^ { 3 } + 10 x ^ { 2 } - 3 a x - 4$$
Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).
You must make your method clear.
- Solutions via this method must end with the value for \(a\) to score the A1
## Question 1:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(1) = a(1)^3 + 10(1)^2 - 3a(1) - 4 = 0$ | M1 | Attempts $f(1) = 0$ to set up an equation in $a$. Implied by $a + 10 - 3a - 4 = 0$. Condone a slip; attempting $f(-1) = 0$ is M0 |
| $6 - 2a = 0 \Rightarrow a = \ldots$ | M1 | Solves a linear equation in $a$. Implied by a solution of $\pm a \pm 10 \pm 3a \pm 4 = 0$. Don't be concerned about mechanics of solution |
| $a = 3$ | A1 | Following correct work |
**Total: 3 marks**
**Additional Notes:**
- Answers without working score 0 marks
- If candidate states $a = 3$ and shows $f(x) = 3x^3 + 10x^2 - 9x - 4$ with $(x-1)$ as a factor by an allowable method, award M1 M1 A1
- E.g. 1: $3x^3 + 10x^2 - 9x - 4 = (x-1)(3x^2 + 13x + 4)$, hence $a = 3$
- E.g. 2: $f(x) = 3x^3 + 10x^2 - 9x - 4$, $f(1) = 3 + 10 - 9 - 4 = 0$, hence $a = 3$
- Solutions via this method must end with the value for $a$ to score the A1
1.
$$f ( x ) = a x ^ { 3 } + 10 x ^ { 2 } - 3 a x - 4$$
Given that $( x - 1 )$ is a factor of $\mathrm { f } ( x )$, find the value of the constant $a$.\\
You must make your method clear.
\hfill \mbox{\textit{Edexcel Paper 1 2021 Q1 [3]}}