If f(x)=x^4-2 x^3+3 x^2-a x+b is divided by x-1 and x+1 , the remainders are 5 and 19, respectively. If f(x) is divided by x-2 . The remainder is

Step-by-step solution

f(x)=x^4-2 x^3+3 x^2-a x+b Given that, f(1)=5 and f(-1)=19 , then aligned & f(1)=1-2+3-a+b=2-a+b=5 & b-a=3 & and f(-1)=1+2+3+a+b & =b+a+6=19 & b+a=13 & aligned adding Eqs. (i) and (ii), we obtain aligned & 2 b=16 b=8 & and a=13-b a=5 & f(x)=x^4-2 x^3+3 x^2-5 x+8 aligned Thus, aligned f(2) & =(2)^4-2(2)^3+3(2)^2-5(2)+8 & =16-16+12-10+8 & =10 aligned If f(x) is divided by x-2 , then remainder is 10 .