- In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
- The quartic equation
$$z ^ { 4 } + 5 z ^ { 2 } - 30 = 0$$
has roots \(p , q , r\) and \(s\).
Without solving the equation, determine the quartic equation whose roots are
$$( 3 p - 1 ) , ( 3 q - 1 ) , ( 3 r - 1 ) \text { and } ( 3 s - 1 )$$
Give your answer in the form \(w ^ { 4 } + a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be found. - The roots of the cubic equation
$$4 x ^ { 3 } + n x + 81 = 0 \quad \text { where } n \text { is a real constant }$$
are \(\alpha , 2 \alpha\) and \(\alpha - \beta\)
Determine
(a) the values of the roots of the equation,
(b) the value of \(n\).