Given that a cubic equation has three distinct roots that all lie on the same straight line in the complex plane,
describe the possible lines the roots can lie on.
$$f ( z ) = 8 z ^ { 3 } + b z ^ { 2 } + c z + d$$
where \(b , c\) and \(d\) are real constants.
The roots of \(f ( z )\) are distinct and lie on a straight line in the complex plane.
Given that one of the roots is \(\frac { 3 } { 2 } + \frac { 3 } { 2 } \mathrm { i }\)
state the other two roots of \(\mathrm { f } ( \mathrm { z } )\)
$$g ( z ) = z ^ { 3 } + P z ^ { 2 } + Q z + 12$$
where \(P\) and \(Q\) are real constants, has 3 distinct roots.
The roots of \(g ( z )\) lie on a different straight line in the complex plane than the roots of \(\mathrm { f } ( \mathrm { z } )\)
Given that
\(f ( z )\) and \(g ( z )\) have one root in common
one of the roots of \(\mathrm { g } ( \mathrm { z } )\) is - 4
write down the value of the common root,
determine the value of the other root of \(\mathrm { g } ( \mathrm { z } )\)
Hence solve the equation \(\mathrm { f } ( \mathrm { z } ) = \mathrm { g } ( \mathrm { z } )\)