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

"Solution
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`
and `(-1)^4-2×(-1)³ + 3 × (−1)² − a × (-1) + b = 19`
⇒`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)=x²-2x³ + 3x² − 5x+8`The remainder when `f(x)` is divided by `(x-2)` is equal to `f(2)`.
So, Remainder = `f(2)= (2)¹ − 2 × (2)³ + 3×(2)²-5×2+8=16-16+12 - 10+8 = 10`"