2020 · Q22
If f(x) is divided by (x – α)(x – β) where α ≠ β, then what is the remainder?
Elementary MathematicsAlgebra
- A((x - α)f(α) - (x - β)f(β))/(α - β)
- B((x - α)f(β) - (x - β)f(α))/(α - β)
- C((x - β)f(α) - (x - α)f(β))/(α - β)
- D((x - β)f(β) - (x - α)f(α))/(α - β)
Explanation
The remainder on division by a quadratic is linear: R(x) = px + q with R(α) = f(α) and R(β) = f(β). Lagrange interpolation gives R(x) = f(α)(x − β)/(α − β) + f(β)(x − α)/(β − α) = [(x − β)f(α) − (x − α)f(β)]/(α − β). This is the Lagrange interpolation form of the linear remainder.