For $n$ a positive whole number, and for $x \neq 0$, let $p _ { n } ( x ) = x ^ { n } + x ^ { - n }$. [0pt] (i) [3 marks] Sketch the graph of $y = p _ { 1 } ( x )$. Label any turning points on your sketch. (ii) $[ 1$ mark $]$ Show that $p _ { 2 } ( x ) = p _ { 1 } ( x ) ^ { 2 } - 2$. (iii) $[ 1$ mark $]$ Find an expression for $p _ { 3 } ( x )$ in terms of $p _ { 1 } ( x )$. [0pt] (iv) [5 marks] Find all real solutions $x$ to the equation $$x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } + x + 1 = 0$$ (v) [5 marks] Find all real solutions $x$ to the equation $$x ^ { 7 } + 2 x ^ { 6 } - 5 x ^ { 5 } - 7 x ^ { 4 } + 7 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1 = 0 .$$
\section*{2. For ALL APPLICANTS.}
For $n$ a positive whole number, and for $x \neq 0$, let $p _ { n } ( x ) = x ^ { n } + x ^ { - n }$.\\[0pt]
(i) [3 marks] Sketch the graph of $y = p _ { 1 } ( x )$. Label any turning points on your sketch.\\
(ii) $[ 1$ mark $]$ Show that $p _ { 2 } ( x ) = p _ { 1 } ( x ) ^ { 2 } - 2$.\\
(iii) $[ 1$ mark $]$ Find an expression for $p _ { 3 } ( x )$ in terms of $p _ { 1 } ( x )$.\\[0pt]
(iv) [5 marks] Find all real solutions $x$ to the equation
$$x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } + x + 1 = 0$$
(v) [5 marks] Find all real solutions $x$ to the equation
$$x ^ { 7 } + 2 x ^ { 6 } - 5 x ^ { 5 } - 7 x ^ { 4 } + 7 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1 = 0 .$$