Question

Which of the following are roots of the following polynomial x^3-11x^2+33x+45

A. -3, 6+i, 6-i
B. -1, 6+3i, 6-3i
C. -3, 5+i, 5-i
D. -1, 5+i, 5-i

Answer 1

For x^3-11x^2+33x+45 , we can make it an equation so x^3-11x^2+33x+45=0. Next, we can find out if -1 or -3 is a factor. If -1 is a factor, than (x+1) is factorable. Using synthetic division, we get

x^2-12x+45
___ ________________________
x+1 | x^3-11x^2+33x+45

- (x^3+x^2)
_________________________
-12x^2+33x+45
- (-12x^2-12x)
______________
45x+45
-(45x+45)
___________
0


Since that works, it's either B or D. We just have to figure out when
x^2-12x+45 equals 0, since there are 3 roots and we already found one. Using the quadratic formula, we end up getting (12+-sqrt(144-180))/2=
(12+-sqrt (-36))/2. Since sqrt(-36) is 6i, and 6i/2=3i, it's pretty clear that B is our answer