Standard +0.8 This is a Further Maths question requiring students to use Vieta's formulas with the special relationship between roots (α and 1/α are reciprocals). While the technique is standard for FM students, it requires careful algebraic manipulation across multiple steps: using the product of roots to find α²β = -3, the sum to get a relationship involving k, and the sum of products to establish k. The exact form requirement adds modest complexity. This is moderately challenging but well within expected FM Pure content.
4 In this question you must show detailed reasoning.
The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.
4 In this question you must show detailed reasoning.
The equation $z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0$, where $k$ is a constant, has roots $\alpha , \frac { 1 } { \alpha }$ and $\beta$.\\
Determine the roots in exact form.
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2022 Q4 [6]}}