Answer:
Since we know that complex solutions ALWAYS come in pairs, the minimal polynomial must include the root of 4-i as an acceptable root..
This leads to a polynomial of ...
p(x) = (x - 5)(x - (4+i))(x - (4-i))
Now we can simplify...
(x - (4+i))(x - (4-i)) --> x2 + [(4+i)+(4-i)]x + (4+i)(4-i) --> x2 + 8x + [ 16 -4i + 4i -i2 ] = x2 + 8x +17
Take this quadratic and multiply by (x-5) to get the final, minimal polynomial...
p(x) = (x-5)(x2 + 8x +17) = x3 + 8x2 + 17x -5x2 + 40x - 85 = x3 + 3x2 + 57x - 85
So p(x) = x3 + 3x2 + 57x - 85
p(x) has only real coefficients, the leading coefficient of 1, and is the smallest to allow for the two solutions requested
