Question

give me proper answer​

Answer 1

Given---> If ( x + a ) is a factor of the polynomial

( x² + px + q ) and ( x² + mx + n )

To prove---> a = ( n - q ) / ( m - p )

Proof---> Let, P ( x ) = x² + px + q and

Q( x ) = x² + mx + n and g ( x ) = x + a

Now , ATQ, g ( x ) is factor of P( x ) and Q ( x ) .

Now , putting , g ( x ) = 0

=> x + a = 0

=> x = - a

Now, g ( x ) is a factor of P ( x ) , so by factor theorem,

P ( - a ) = 0

=> ( - a )² + p ( - a ) + q = 0

=> a² - ap + q = 0 ..........................( 1 )

Now , g ( x ) is a factor of Q ( x ) so by factor theorem,

Q ( - a ) = 0

=> ( - a )² + m ( - a ) + n = 0

=> a² - a m + n = 0 ........................( 2 )

By (1 ) and ( 2 ) , we get,

=> a² - ap + q = a² - am + n

=> - ap + q = - am + n

=> am - ap = n - q

=> a ( m - p ) = n - q

=> a = ( n - q ) / ( m - p )