1) The polynomial f(x) is divided by the polynomial d(x) to give a quotient of q(x) and a remainder of r(x). If deg f = 9 and deg r = 3, what is the maximum possible value of deg q?
2) When the polynomial p(x) is divided by x - 1, the remainder is 3. When the polynomial p(x) is divided by x - 3, the remainder is 5. What is the remainder when the polynomial p(x) is divided by (x - 1)(x - 3)?
+9519 1)
\(f(x) = q(x) d(x) + r(x)\\ \text{If }\text{deg}\;d \le 3, \text{then deg } r < \text{deg }d \le 3 \implies \text{Contradiction!}\\ \therefore \text{deg }d \ge 4\\ \text{Minimum possible deg }d = 4\\ \implies \text{Maximum possible deg }q = \text{deg }f - \text{deg }d = \boxed5\)
