Question

23.
Letf(x) be a polynomial leaving the remainder A when divided by (x -a) and remainder B when divided by x=b,
Find remainder left by this polynomial when divided by (x - a) (x-b).​

Answer 1

Answer:

The remainder is

( ( A - B ) / ( a - b ) ) x  +  ( aB - bA ) / ( a - b )

Step-by-step explanation:

Since (x-a)(x-b) has degree two, the remainder has degree at most 1, so it has the form cx+d.  We need to work out c and d.

Since cx+d is the remainder when dividing f(x) by (x-a)(x-b), this means that for some other polynomial g(x) we have

 f(x) = (x-a)(x-b)g(x) + cx + d    ... (1)

By the remainder theorem, A=f(a) and B=f(b).  Together with (1) above, this gives

 ac + d = A     ... (2)

 bc + d = B     ... (3)

Subtracting (3) from (2) gives

 ( a - b ) c = A - B  ⇒ c = ( A - B ) / ( a - b )

Subtracting b times (2) from a times (3) gives

 ( a - b ) d = aB - bA  ⇒  d = ( aB - bA ) / ( a - b )

Hope this helps!