Question

Let p(x) be a polynomial more than 2. when p(x) is divided by x - 2 it leaves remainder 1 and when it is divided by x - 3, it leaves remainder, when p(x) is divided by (x-2) (x-3)

Answer 1

Given, p(x) is a polynomial of degree more than 2. 

when p(x) is divided by (x-2)(x-3), then,

p(x)=(x-2)(x-3)g(x) + (ax+b).......(1)

We know that, if the polynomial p(x) is divided by (x-a), then the remainder is p(a).

Given, when p(x) is divided by (x-2), it leaves remainder 1.

So p(2)=1
1=(2-2)(2-3)g(2)+(2a+b)........ from 1

And, when p(x) is divided by (x-3), it leaves remainder 3.


so p(3)= 3
3=(3-2(3-3)g(3)+(3a+b).......from1


Solving equation (2) and (3), we have,


a=2 b= -3

So, equation (1) becomes,

p(x)= (x-2)(x-3)(gx)+(2x-3)

Hence, when p(x) is divided by (x-2)(x-3), then remainder will be (2x-3).


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