<s>[INST] If $a,b$ are the roots of the equation $x^2-2x+3=0$ obtain the equation whose roots are $a^3-3a^2+5a-2$, $b^3-b^2+b+5$ 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.
 [/INST] 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 
$$ (X-1)(X-2)=X^2-3X+2.$$
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