Final answer:
To solve the problem, use the Remainder Theorem and set up equations to find the values of a and b. The values of a and b are 7 and 0, respectively.
Step-by-step explanation:
To solve this problem, we can use the Remainder Theorem. When we divide the polynomial f(z) by z-i, the remainder is 8. This means that f(i) = 8. Similarly, when we divide f(z) by z-a, the remainder is 1-i, which means that f(a) = 1-i. Since f(i) = 8, we can substitute z=i into the polynomial f(z) to get f(i) = a(i-i)+b=(a-8)+(b-8i)=8.
Similarly, since f(a) = 1-i, we can substitute z=a into the polynomial f(z) to get f(a) = a(a-a)+b=(a-1)+(b-i)=1-i.
Setting these two equations equal to each other, we get (a-8)+(b-8i) = (a-1)+(b-i).
Expanding and rearranging, we find that -7 + (b-8i) = -1 + (b-i). This implies that -7 = -1 and -8i = -i. Simplifying, we find that a = 7 and b = 0.