Question 390785
There's an important theorem called "The Fundamental Theorem of Algebra" (it looks easy, but requires high-level analysis to prove) that says that any n-degree polynomial has n roots, counting multiplicities.


Furthermore, if r is a root of the polynomial, then (x-r) is a factor of it (this will be useful in generating the polynomial).


Therefore the polynomial P(x) is equal to


{{{P(x) = (x-3/4)(x+1/3)(x - a - bi)(x - a - bi)}}}


You write in your problem "Then, create the list of all possible rational roots and use synthetic division to reduce the 4th degree polynomial to a quadratic and solve for the remaining roots. Show all steps." However, the only rational roots are 3/4 and -1/3. The quadratic with a double root x - a - bi has double root {{{a + bi}}}. Therefore, solving for the remaining roots using synthetic division is unnecessary...only do that if you're practicing your synthetic division :)