a) In this multi-part problem, we will consider this system of simultaneous equations:
\(\begin{array}{r@{~}c@{~}l l} 3x+5y-6z &=&2, & \textrm{(i)} \\ 5xy-10yz-6xz &=& -41, & \textrm{(ii)} \\ xyz&=&6. & \textrm{(iii)} \end{array}\)
Let \(a=3x, b=5y,\) and \(c=-6z\).
Determine the monic cubic polynomial in terms of a variable \(t\) whose roots are \(t=a,t=b,\) and \(t=c\). Make sure to enter your answer in terms of \(t\) and only \(t\), in expanded form.
(Suggestion: As a first step, you may wish to rewrite the system of equations in terms of \(a,b,\) and \(c\).)
b) Given that \((x,y,z)\) is a solution to the original system of equations, determine all distinct possible values of \(x+y\).
(Suggestion: Using the substitutions in part (a), first determine all possible values of the ordered triple \((a,b,c)\), then determine the possible solutions \((x,y,z)\).)
thanks in advance!
