A2 June 2024 Paper 2 Q11
11 Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots.
Latifa says that if such an equation has exactly one real root, it must be of degree 3
Sam says that this is not correct.
State, giving reasons, whether Latifa or Sam is right. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Refers to polynomials of odd and/or even degree. Or States that a polynomial of a particular odd degree greater than 3 can have exactly one real root or sketches a graph to show this. or Obtains a polynomial of degree greater than three with exactly one real root. | M1 | 2.4 |
| Explains that complex roots occur in conjugate pairs (condone “imaginary roots”). or Explains that their specific polynomial is a counter example to Latifa’s statement. | M1 | 2.4 |
| Completes a reasoned argument to conclude that Sam is right (do not condone “imaginary roots”) and States clearly that Sam is right. OE | R1 | 2.3 |
| (3 marks) |
Typical solution
The polynomial equation \(z^5 - 1 = 0\) is of degree 5 and has exactly one real root.
This is a counter example to Latifa’s statement.
So Sam is right.