已知Y的2次方+3Y-1=0求Y的8次方-3Y+1的值

已知Y的2次方+3Y-1=0求Y的8次方-3Y+1的值
已知Y的2次方+3Y-1=0求Y的8次方-3Y+1的值

已知Y的2次方+3Y-1=0求Y的8次方-3Y+1的值
已知,Y^2+3Y-1 = 0 ,解得：Y = (-3±√13)/2 .
第一种做法：
Y^2 = -3Y+1 ；
Y^4 = (-3Y+1)^2 = 9Y^2-6Y+1 = 9(-3Y+1)-6Y+1 = -33Y+10 ；
Y^8 = (-33Y+10)^2 = 1089Y^2-660Y+100 = 1089(-3Y+1)-660Y+100 = -3927Y+1189 ；
Y^8-3Y+1 = -3927Y+1189-3Y+1 = -3930Y+1190 = 7085±1985√13 .
第二种做法：
Y^8-3Y+1 = Y^8+Y^2 = Y^2(Y^6+1) = Y^2(Y^2+1)(Y^4-Y^2+1)
= Y^2(Y^2+1)[Y^2(Y^2-1)+1] = (-3Y+1)(-3Y+2)[(-3Y+1)(-3Y)+1]
= (9Y^2-9Y+2)(9Y^2-3Y+1) = (-27Y+9-9Y+2)(-27Y+9-3Y+1)
= (-36Y+11)(-30Y+10) = 1080Y^2-690Y+110
= -3240Y+1080-690Y+110 = -3930Y+1190
= 7085±1985√13 .