Given that  is a root of the equation  If  where  are co-primes, find 

Asked by 5324038.08.anthony | 12th Jun, 2021, 11:26: AM

Expert Answer:

As 'a' is a root of the given equation, we have
a2 - a - 3 = 0
i.e. a2 - a = 3
Consider comma
fraction numerator straight a cubed plus 1 over denominator straight a to the power of 5 minus straight a to the power of 4 minus straight a cubed plus straight a squared end fraction
equals fraction numerator open parentheses straight a plus 1 close parentheses open parentheses straight a squared minus straight a plus 1 close parentheses over denominator straight a to the power of 4 open parentheses straight a minus 1 close parentheses minus straight a squared open parentheses straight a minus 1 close parentheses end fraction
equals fraction numerator open parentheses straight a plus 1 close parentheses open parentheses straight a squared minus straight a plus 1 close parentheses over denominator open parentheses straight a to the power of 4 minus straight a squared close parentheses open parentheses straight a minus 1 close parentheses end fraction
equals fraction numerator open parentheses straight a plus 1 close parentheses open parentheses straight a squared minus straight a plus 1 close parentheses over denominator straight a squared open parentheses straight a squared minus 1 close parentheses open parentheses straight a minus 1 close parentheses end fraction
equals fraction numerator open parentheses straight a plus 1 close parentheses open parentheses straight a squared minus straight a plus 1 close parentheses over denominator straight a squared open parentheses straight a plus 1 close parentheses open parentheses straight a minus 1 close parentheses open parentheses straight a minus 1 close parentheses end fraction
equals fraction numerator open parentheses straight a squared minus straight a plus 1 close parentheses over denominator straight a squared open parentheses straight a minus 1 close parentheses squared end fraction
equals fraction numerator open parentheses straight a squared minus straight a plus 1 close parentheses over denominator open parentheses straight a squared minus straight a close parentheses squared end fraction
equals fraction numerator 3 plus 1 over denominator 3 squared end fraction
equals 4 over 9
Therefore comma space straight p equals 4 space and space straight q space equals space 9
Thus comma space straight p plus straight q space equals space 4 plus 9 space equals space 13

Answered by Renu Varma | 14th Jun, 2021, 11:35: AM