Question

find the(monic) polynomial equation of lowest degree whose roots are 2,3,6​

Answer 1

Answer:

Let y=2–√3. Then x=y+3y2=y(3y+1) so cubing both sides yields

x3=y3(27y3+27y2+9y+1)=2(27⋅2+9y(3y+1)+1)=2(9x+55)

so x3−18x−110=0. This is the minimal polynomial as [Q(2–√3):Q]=3.

Answer 2

Answer:

x3-11x2+36x-36

Step-by-step explanation:

given roots alpha=2 beta=3 and gama=6

s1( alpha+beta+gama)= 2+3+6=11

s2(alpha.beta+beta.gama+gama.alpha)=6+18+12=36

s3(alpha.beta.gama)=36

monic polynomial of degree 3: x3-s1(x2)+s2(x)-s3=0

required equation is x3-11x2+36x-36 = 0