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If the remainders of the polynomial f(x) when divided by x-1,x-2 are 2,5 then the remainder of f(x) when divided by (x-1)( x-2) is

If the remainders of the polynomial f(x) when divided by x-1,x-2 are 2,5 then the remainder of f(x) when divided by (x-1)( x-2) is

Grade:12th pass

1 Answers

Arun
25750 Points
6 years ago
When a polynomial f ( x ) is divided by x - 1 and x - 2, the remainders are 5 and 7 respectively.

So,
We get  f ( x  ) =  ( x  - 1 ) q ( x )  + 5

We substitute x  =  1  , we get 

f (  1 ) =  ( 1  -  1 ) q ( x ) + 5

f (  1 ) =  5

And

f ( x  ) =  ( x -  2 ) Q ( x )  + 5

We substitute x  =  2  , we get 

f (  2 ) =  (  2 - 2 ) Q ( x ) + 5

f (  2 ) =  5

Now Let the remainder Ax + B  When f ( x )  divide by ( x - 1 ) ( x - 2 ) , So

f ( x  ) =  ( x  - 1 ) ( x - 2 )p ( x )  + Ax + B                     ----- (  A )

We substitute x  = 1 , we

f ( 1 ) =  (  1 - 1 ) (  1 - 2 )p ( x )  + A( 1 ) + B

f ( 1 ) =  0  + A  + B  , Substitute value of f (  1 )  we get

A +  B =  2                         ----- ( 1 )

And we substitute x  =   2 in equation A , we get

f (   2 ) =  (  2 - 1 ) (  2  - 2 )p ( x )  + A( 2 ) + B

f ( 2 ) =  0  + 2A  + B  , Substitute value of  f ( 2 )  we get

2A +  B =  5                         ----- ( 2 )

Now we subtract equation 1 from equation 2 , we get

A  = 3 , Substitute that value in equation 1 , we get

3 + B  = 2

B  = – 1 
So,
Remainder  =  Ax + B = ( 3 ) x   – 1  =  3x  –  1

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