Standard +0.8 This requires understanding that complex roots come in conjugate pairs for polynomials with real coefficients, then constructing a counterexample (e.g., degree 5 with one real root and two conjugate pairs). It tests conceptual understanding rather than routine calculation, but the insight needed is within reach of strong Further Maths students who understand the conjugate root theorem.
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]
Question 11:
11 | 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. | 2.4 | M1 | The polynomial equationz5 −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.
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. | 2.4 | M1
Completes a reasoned
argument to conclude that Sam
is right (do not condone
“imaginary roots”)
and
States clearly that Sam is right.
OE | 2.3 | R1
Question total | 3
Q | Marking instructions | AO | Marks | Typical solution
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]
\hfill \mbox{\textit{AQA Further Paper 2 2024 Q11 [3]}}