Consider the quadratic polynomial p(x)=ax2+bx+12.
(i) When p(x) is divided by x−2, the remainder is 6. Write an equation connecting a and b based on this information and clearly explain the theorem that justifies the equation.
(ii) Explain why the information in part (i) alone is not sufficient to determine unique values of both a and b.
(iii) If it is further given that (x+1) is a factor of p(x), form a second equation in a and b (do not solve the system). Describe each reasoning step you use.