3.1: Parameter from a linear factor, then unique real root

0606/12/F/M/25 — Question 4 · 6 marks

The polynomial pp is given by p(x)=a2x3+2ax2+ax+2p(x) = a^{2}x^{3} + 2ax^{2} + ax + 2, where aa is a positive integer.
It is given that 2x+12x + 1 is a factor of p(x)p(x).
(a)
Find the value of aa.
[3]
(b)
Hence factorise p(x)p(x).
[2]
(c)
Hence show that the equation p(x)=0p(x) = 0 has only one real root.
[1]