If $a,b$ are the roots of the equation $x^2-2x+3=0$ obtain the equation whose roots are $a^3-3a^2+5a-2$, $b^3-b^2+b+5$

Question

I have been trying this using sum of roots and product of roots but it gets too lengthy. So I found the roots of the given equation which are imaginary and tried to replace the values in the two given roots. Still I am not able to solve this.

Answer 1

We have $$ a^3-3a^2+5a-2=(a-1)(a^2-2a+3)+1=1$$ and $$b^3-b^2+b+5=(b+1)(b^2-2b+3)+2=2. $$ So the desired polynomial is $$ (X-1)(X-2)=X^2-3X+2.$$

Answer 2

The problem is not very well formulated: if you want a degree-two polynomial with roots $\alpha$ and $\beta$, just take $$ (x-\alpha)(x-\beta). $$ The way the problem is phrased it is clear that something else is expected, but I cannot imagine what.