Question

if a polynomial is divided by (x-2) itr leaves remainder 1. if it is divided by (x-3) it leaves remainder 3.what will be the remainder of the same polynomial when divided by (x-2)(x-3)

Answer 1

Answer :
Let our polynomial = f ( x )

By remainder theorem , we know

f ( 2 ) = 1 ( As remainder ) ------------------ ( 1 )
And
f ( 3 ) = 3 ( As remainder ) ------------------ ( 1 )

And let our remainder = ax + b when polynomial divide by ( x - 2 ) ( x - 3 )
So,
f ( x ) = q ( x ) ( x - 2 ) ( x - 3 ) + ax + b

At x = 2 , we get

$\Rightarrow$f ( 2 ) = q ( 2 ) ( 2 - 2 ) ( 2 - 3 ) + 2a + b

Substitute value of f ( 2 ) from equation 1 , and get

2a + b = 1 ------------------- ( 3 )

Now ​At x = 3 , we get

$\Rightarrow$f ( 3 ) = q ( 3 ) ( 3 - 2 ) ( 3 - 3 ) + 3a + b

Substitute value of f ( 3 ) from equation 2 , and get

3a + b = 3 ------------------- ( 4 )

Now we subtract equation 3 from equation 4 ,and get

$\Rightarrow$a = 2 ( substitute that value in equation 3 , we get

$\Rightarrow$2 ( 2 ) + b = 1

$\Rightarrow$b = 1 - 4

$\Rightarrow$b = - 3

SO,

Our remainder will be = 2x - 3 ( Ans )