Remainder when f(x) is divided by x - 3

To find the remainder when f(x) is divided by x - 3, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial f(x) is divided by x - c, the remainder is equal to f(c).

Given that the remainder when f(x) is divided by (x - 1) is 5, we have f(1) = 5. Similarly, the remainder when f(x) is divided by (x + 1) is 19, so f(-1) = 19.

Using the Remainder Theorem:
To find the remainder when f(x) is divided by x - 3, we need to evaluate f(3).

So, let's substitute x = 3 into the polynomial f(x) = x^4 - 2x^3 + 3x^2 - ax + b:

f(3) = (3)^4 - 2(3)^3 + 3(3)^2 - a(3) + b

Simplifying further:

f(3) = 81 - 54 + 27 - 3a + b
f(3) = 54 - 3a + b

Using the given remainders:
We know that f(1) = 5 and f(-1) = 19. Let's substitute these values into the equation we obtained for f(3):

54 - 3a + b = 5 ... (1)
54 - 3a + b = 19 ... (2)

Solving the equations:
Now, we have two equations with two unknowns (a and b). We can solve these equations simultaneously to find the values of a and b.

Subtracting equation (2) from equation (1), we get:

5 - 19 = 54 - 54 - 3a + 3a + b - b
-14 = 0

The equation -14 = 0 is not possible, meaning there is no solution for this system of equations. This implies that there is no unique value for the remainder when f(x) is divided by x - 3.

Therefore, the remainder when f(x) is divided by x - 3 is indeterminate or cannot be determined.

Note: It is important to double-check the given information to ensure that there are no errors or inconsistencies in the problem statement.