a and b are integers such that the polynomial f(x)=x^4−4x^3−8x^2+ax+b has 1+2i as a solution to f(x)=0. What is the value of 10a+b?

Answer 1

another root is also 1-2i, so

(x-1)^2+4 = x^2-2x+5 is a divisor of f(x)

(x^2-2x+5)(x^2+mx+n)
= x^4 + (m-2)x^3 + (n-2m+5)x^2 + (5m-2n)x + (5n)

So,

m-2 = -4
n-2m+5 = -8

a = 5m-2n
b = 5n

m = -2
n = -17

a = 24
b = -85

10a+b = 155

check:
(x^2-2x+5)(x^2-2x-17) = x^4−4x^3−8x^2+24x-85