P(x) is a polynomial of degree more than 2 when p(x) is divided by x-2 it leaves a remainder 1 and when it is divided by x-3 it leaves a remainder 3. Find the remainder when p)x) is divided by(x-2)(x-3) ?

1 Answer
Jun 24, 2018

The remainder is =x

Explanation:

By the remainder theorem

When the polynomial P(x) is divided by (x-2), the remainder is =2

=>, P(2)=2

When the polynomial is divided by (x-3), the remainder is =3

=>, P(3)=3

Let D=(x-2)(x-3)

When the polynomial is divided by (x-2)(x-3), the quotient is Q and the remainder R will be of the form Ax+B

Therefore,

P(x)=QD+Ax+B

So,

P(2)=Q*0+2A+B

=>, 2A+B=2...............(1)

P(3)=Q*0+3A+B

=>, 3A+B=3....................(2)

Solving equations (1) and (2) for A and B

{(2A+B=2),(3A+B=3):}

=>, {(2A+B=2),(A=1):}

=>, {(A=1),(B=0):}

Therefore,

The remainder is =x