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 8. What is the remainder when the polynomial p(x) is divided by (x - 1)(x - 3)?
+526 According to remainder theorem
When a polynomial p(x) is divided by (x−a), the remainder is p(a).
When p(x) divided by x-1, remainder is 3
... p(1) = 3
When p(x) divided by x-3, remainder is 8
... p(3) = 8
When p(x) is divided by (x-1)(x-3) let remainder be r(x) = ax + b
⇒ p(x) = (x-1)(x-3)q(x) + ax + b
p(1) = a + b = 3
p(3) = 3a + b = 8
Subtracting p(1) from p(3),
2a = 5 ⇒ a = 5/2
⇒ b = 1/2
Thus the remainder is \(r(x) = {5\over 2}x+{1\over 2}\)
