How do you find a fourth degree polynomial given roots #2+i# and #1-sqrt5#?

1 Answer
Feb 15, 2017

Answer:

#x^4 -6x^3 +10x^2 +2x -15#

Explanation:

If 2+i is a root, its conjugate 2-i would also be a root. Likewise, if #1-sqrt5# is a root, its conjugate #1+sqrt5# would also be root. The polynomial would thus be

#(x -2-i)(x-2+i)(x-1+sqrt5)(x-1-sqrt5)#

#=((x-2)^2 +1) ((x-1)^2 -5)#

#=(x^2 -4x +5)(x^2 -2x -3)#

#=x^4 -6x^3 +10x^2 +2x -15#