There is a unique real number $\alpha$ that satisfies the equation $$\alpha ^ { 3 } + \alpha ^ { 2 } = 1$$ [You are not asked to prove this.] (i) Show that $0 < \alpha < 1$. (ii) Show that $$\alpha ^ { 4 } = - 1 + \alpha + \alpha ^ { 2 }$$ (iii) Four functions of $\alpha$ are given in (a) to (d) below. In a similar manner to part (ii), each is equal to a quadratic expression $$A + B \alpha + C \alpha ^ { 2 }$$ in $\alpha$, where $A , B , C$ are integers. (So in (ii) we found $A = - 1 , B = 1 , C = 1$.) You may assume in each case that the quadratic expression is unique. In each case below find the quadratic expression in $\alpha$. (a) $\alpha ^ { - 1 }$. (b) The infinite sum $$1 - \alpha + \alpha ^ { 2 } - \alpha ^ { 3 } + \alpha ^ { 4 } - \alpha ^ { 5 } + \cdots$$ (c) $( 1 - \alpha ) ^ { - 1 }$. (d) The infinite product $$( 1 + \alpha ) \left( 1 + \alpha ^ { 2 } \right) \left( 1 + \alpha ^ { 4 } \right) \left( 1 + \alpha ^ { 8 } \right) \left( 1 + \alpha ^ { 16 } \right) \cdots$$
(i) [3 marks] We have $\alpha ^ { 2 } ( 1 + \alpha ) = 1$. Now:
\section*{2. For ALL APPLICANTS.}
There is a unique real number $\alpha$ that satisfies the equation
$$\alpha ^ { 3 } + \alpha ^ { 2 } = 1$$
[You are not asked to prove this.]\\
(i) Show that $0 < \alpha < 1$.\\
(ii) Show that
$$\alpha ^ { 4 } = - 1 + \alpha + \alpha ^ { 2 }$$
(iii) Four functions of $\alpha$ are given in (a) to (d) below. In a similar manner to part (ii), each is equal to a quadratic expression
$$A + B \alpha + C \alpha ^ { 2 }$$
in $\alpha$, where $A , B , C$ are integers. (So in (ii) we found $A = - 1 , B = 1 , C = 1$.) You may assume in each case that the quadratic expression is unique.
In each case below find the quadratic expression in $\alpha$.\\
(a) $\alpha ^ { - 1 }$.\\
(b) The infinite sum
$$1 - \alpha + \alpha ^ { 2 } - \alpha ^ { 3 } + \alpha ^ { 4 } - \alpha ^ { 5 } + \cdots$$
(c) $( 1 - \alpha ) ^ { - 1 }$.\\
(d) The infinite product
$$( 1 + \alpha ) \left( 1 + \alpha ^ { 2 } \right) \left( 1 + \alpha ^ { 4 } \right) \left( 1 + \alpha ^ { 8 } \right) \left( 1 + \alpha ^ { 16 } \right) \cdots$$