2025 · Q1
Let p(x) be a polynomial. When p(x) is divided by (x-1), it leaves 2 as the remainder. When p(x) is divided by (x-2), it leaves 1 as the remainder. What is the remainder when p(x) is divided by (x-1)(x-2)?
Elementary MathematicsAlgebra
- A3
- B-3
- C3-x
- D3-2x
Explanation
By the Remainder Theorem, p(1)=2 and p(2)=1. The remainder on division by the quadratic (x-1)(x-2) must be linear, say ax+b. Then a+b=2 and 2a+b=1, giving a=-1, b=3. Hence the remainder is 3-x. A common error is to add the two remainders (3) or to assume the remainder is constant.