Question 3 - A-Level Maths

CAIE FP1 2004 November — Question 3 6 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2004
Session	November
Marks	6
Paper	Download PDF ↗
Topic	Roots of polynomials
Type	Finding polynomial from root properties
Difficulty	Standard +0.8 This is a Further Maths question requiring systematic application of Newton's identities to convert power sums into elementary symmetric functions, then constructing the cubic equation and solving it. While the technique is standard for FP1, it requires careful algebraic manipulation across multiple steps and knowledge of the relationship between symmetric functions and polynomial coefficients, making it moderately challenging but within expected Further Maths scope.
Spec	4.05a Roots and coefficients: symmetric functions

3 Given that $$\alpha + \beta + \gamma = 0 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 14 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 18$$ find a cubic equation whose roots are $\alpha , \beta , \gamma$. Hence find possible values for $\alpha , \beta , \gamma$.

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3 Given that

$$\alpha + \beta + \gamma = 0 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 14 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 18$$

find a cubic equation whose roots are $\alpha , \beta , \gamma$.

Hence find possible values for $\alpha , \beta , \gamma$.

\hfill \mbox{\textit{CAIE FP1 2004 Q3 [6]}}

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