If `f(x)=x^4-2x^3+3x^2-a x+b` is `a` polynomial such that when it is divided by `x-1` and `x+1,` the remainders are respectively `5` and `19.` Determine the remainder when `f(x)` is divided by `(x-2)dot`

Explanation

It is given that, `p(1)=5 " " "and" p(-1)=19`
`{:(rArr (1)^(4) -2(1)^(3)+3(1)^(2) -a(1)+b=5,|,"Also," (-1)^(4)-2(-1)^(3)+3(-1)^(2)-a(-1)+b=19),(rArr" " 1-2+3-a+b=5,rArr 1+2+3+a+b=19,),(rArr " " -a+b=3......(1),rArr " " a+b=13....(2),):|`
Adding (1) and (2), we get,
`2b=16 implies b=8`
Putting this value of b in (2), we get
a=5
`therefore p(x)=x^(4)-2x^(3)+3x^(2)-5x+8`
`therefore p(2)=(2)^(4)-2(2)^(3)+3(2)^(2)-5(2)+8=10`
So, remainder =10, when p (x) is divided by (x-2)