4.02: Polynomial parameter and roots

0606/11/M/J/26 — Question 5 · 8 marks

\(p(x)=ax^3+4x^2-ax-28\), where \(a\) is positive, and \(p(a)=0\).
(a)
Find \(a\).
[4]
(b)
Express \(p(x)\) as the product of a linear factor and a quadratic factor.
[2]
(c)
Hence show that \(p(x)=0\) has exactly one real root.
[2]