Let (f(x) ) be a polynomial and (a, b ) be distinct real numbers. Then the remainder in the division of (f(x) ) by ((x-a)(x-b) ) is

Step-by-step solution

Let (f(x)=(x-a)(x-b) q(x)+r(x) ) Let (r(x)= x+ [ deg r(x) < deg ). of divisor] ( aligned f(x) & =(x-a)(x-b) q(x)+ x+ f(a) & = a+ (i) f(b) & = b+ (ii) aligned ) Subtract Eqs. (i) from (ii) ( aligned f(a)-f(b) & = (a-b) & = f(a)-f(b) a-b & = f(b)-f(a) b-a aligned ) Put, ( ) in Eq. (i) ( aligned f(a) & = ( f(b)-f(a) b-a ) a+ f(a)[b-a] & =a f(b)-a f(a)+ (b-a) b f(a)-a f(a) & =a f(b)-a f(a)+ (b-a) & = b f(a)-a f(b) (b-a) r(x) & = x+ & = f(b)-f(a) x b-a + b f(a)-a f(b) (b-a) & = x f(b)-x f(a)+b f(a)-a f(b) b-a r(x) & = (