Question

If a polynomial f(x) when divided by (x - a) gives a remainder ri.
and when divided by (x – b) gives a remainder r2, what is the
remainder of dividing f(x) by g(x) = (x - a)(x - b)?​

Answer 1

Answer:

This time, we can write

f(x)=(x−a)(x−b)g(x)+αx+β,

where if f(x) is of degree n,g(x) is of degree n−2. By substituting in x=a and x=b, we have that

f(a)f(b)=αa+β,=αb+β.(1)

By subtracting the second equation from the first, it follows that

f(a)−f(b)=α(a−b),

and hence, as a≠b,

α=f(a)−f(b)a−b.

From (1), it follows that

β=f(a)−αa=f(a)−f(a)−f(b)a−ba,

so that the remainder term is

αx+β=(f(a)−f(b)a−b)x+f(a)−f(a)−f(b)a−ba=f(a)−f(b)a−bx+(a−b)f(a)−a(f(a)−f(b))a−b=f(a)−f(b)a−bx+af(b)−bf(a)a−b.(2)(3)