if f(x) = `x^4-2x^3+3x^2-ax+b ` is a polynomial such that when it is divisible by x-1 and x+1 remainders are 5 and 19 respectively . determine remainder when it is divisible by `x-1` .

Explanation

When `f(x)` is divided by `x-1` and `x+1` the remainder are `5` and `19` respectively.
`∴f(1)=5` and `f(−1)=19`
`⇒(1)^4−2×(1)^3 +3×(1)^2−a×1+b=5`
`⇒1−2+3−a+b=5` and `1+2+3+a+b=19`
`⇒2−a+b=5` and `6+a+b=19`
`⇒−a+b=3` and `a+b=13`
Adding these two equations, we get
`(−a+b)+(a+b)=3+13`
`⇒2b=16`
`⇒b=8`
Putting `b=8` and `−a+b=3`, we get
`−a+8=3`
`⇒-a=−5``⇒a=5`
Putting the values of `a` and `b` in
`f(x)=x^4−2x^3+3x^2−5x+8`
The remainder when `f(x)` is divided by `(x-1)` is equal to `f(1)`.
So, Remainder
`f(1)=(1)^4−2×(1)^3+3×(1)^2−5×1+8`
`=1−2+3−1-5+8=5`
Hence, the remainder when `f(x)` is dovoded by `x-1` is `5`.