Problem: If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is (i=−1):
Answer Choices:
A. 2
B. −2−433i
C. 0
D. −433i
E. −2
Solution:
Let the numbers be x and y. Then x3+y3=(x+y)3−3xy(x+y) ∴x3+y3=13−3(1)(1)=−2
Consider the equation Z2−Z+1=0 with roots Z1=21+i3,Z2=21−i3.
The sum of these roots is 1 and their product is 1 , so that we may take for our numbers the numbers Z1 and Z2. Therefore Z13+Z23=(21+i3)3+(21−i3)3=81+3i3+3i2⋅3+i3⋅33+1−3i3+3i2⋅3−i3⋅33=82−18=−2