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 $(x - 1)(x - 3)$?
Remainder theorem says that:
“When a polynomial 𝑝(𝑥)p(x) is divided by (𝑥−𝑎)(x−a), the remainder is 𝑝(𝑎)p(a).”
Also, when a polynomial 𝑝(𝑥)p(x) is divided by another polynomial 𝑞(𝑥)q(x),the degree of the remainder is at most 11 less than the degree of 𝑞(𝑥)q(x).
Using the remainder theorem, we can write:
𝑝(𝑥)p(x) can be written as:
𝑟(𝑥)r(x) is the remainder polynomial and 𝑄(𝑥)Q(x) is the Quotient polynomial. Since 𝑟(𝑥)r(x) is linear, 𝑟(𝑥)=𝐴𝑥+𝐵r(x)=Ax+B, where 𝐴A and 𝐵B are arbitrary constants.
Solving the 22 equations, we get 𝐴=1A=1 and 𝐵=2B=2.