Find all the zeros of f(x)=x^3+6x^2−11x+40 given that 1−2i is a zero.

Question

x=

Answer 1

imaginary roots come in pairs

the conjugate, 1+2i, is also a root

x=1-2i and x=1+2i

x-1+2i=0 and x-1-2i=0

multiply the left sides together

([x-1]+2i)([x-1]-2i)=[x-1]^2-4i^2 where i^2=-1

this gives x^2-2x+1+4=x^2-2x+5

divide x^2-2x+5 into the original equation and you get x+8, therefore x=-8 is the third root