Write cubic polynomial whose zeroes are 2+√5/2, 2-√5/2 and 4

Asked by mkmishra101 | 11th Jun, 2016, 03:48: PM

Expert Answer:

begin mathsize 12px style s u m space o f space z e r o e s equals open parentheses 2 plus fraction numerator square root of 5 over denominator 2 end fraction close parentheses plus open parentheses 2 minus fraction numerator square root of 5 over denominator 2 end fraction close parentheses plus 4 equals 8
S u m space t a k i n g space p r o d u c t space o f space t w o space z e r o e s space a t space a space t i m e
equals open parentheses 2 plus fraction numerator square root of 5 over denominator 2 end fraction close parentheses open parentheses 2 minus fraction numerator square root of 5 over denominator 2 end fraction close parentheses plus open parentheses 2 plus fraction numerator square root of 5 over denominator 2 end fraction close parentheses 4 plus open parentheses 2 minus fraction numerator square root of 5 over denominator 2 end fraction close parentheses 4
equals open parentheses 4 minus 5 over 4 close parentheses plus 8 plus 2 square root of 5 plus 8 minus 2 square root of 5
equals 20 minus 5 over 4 equals 75 over 4
P r o d u c t space o f space a l l space z e r o e s
equals open parentheses 2 plus fraction numerator square root of 5 over denominator 2 end fraction close parentheses open parentheses 2 minus fraction numerator square root of 5 over denominator 2 end fraction close parentheses 4
equals open parentheses 4 minus 5 over 4 close parentheses 4
equals 16 minus 5 equals 11

S o space t h e space c u b i c space p o l y n o m i a l space c a n space b e
x cubed minus open parentheses s u m space o f space a l l space z e r o e s close parentheses x squared plus open parentheses s u m space o f space p r o d u c t space o f space t w o space z e r o e s space a t space a space t i m e close parentheses x minus p r o d u c t space o f space a l l space z e r o e s
equals x cubed minus 8 x squared plus 75 over 4 x minus 11 end style

Answered by satyajit samal | 13th Jun, 2016, 09:45: AM