Question

Consider the equation:
X^3 - (1 + cosө + sinө) x^2 – ( cosөsinө + cosө + sinө)x – cosөsinө = 0, whose roots are a,b,c.

1. Find a^2 + b^2 + c^2.
2. Number of values of ө є [0,2π] such that the equation has at least two equal roots.
3. The greatest possible difference between the roots.

Answer 1

well looking at your problem knowing the roots your polynomial

P(x) = (x -a)(x-b)(x -c)

if you expand distribute to get you polynomial you will be able to compare the coefficients of each term

given

\[P(x) = x^3 -(1 \cos(\theta) + \sin(\theta)x^2 -(\cos(\theta)\sin(\theta)+\cos(\theta)+\sin(\theta))x - \cos(\theta)\sin(\theta)\]

and looking at your 1st problem

\[(a + b + c)(a + b + c) = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\]

so you need to manipulate the equation so that you are looking at

\[(a + b + c)^2 - something\]

hope this makes sense