The expression ax^2 + bx + c is divisible by x - 1, has a remainder of 2 when divided by x - 1, and has a remainder of 8 when divided by x - 2. Find the values of a, b, and c.

To find the values of a, b, and c, we can use the remainder theorem.

Since the expression is divisible by x - 1, we know that when x = 1, the expression equals 0. This gives us the equation:

a(1)^2 + b(1) + c = 0
a + b + c = 0

Since the expression has a remainder of 2 when divided by x - 1, we know that when x = 1, the expression equals 2. This gives us the equation:

a(1)^2 + b(1) + c = 2
a + b + c = 2

Since the expression has a remainder of 8 when divided by x - 2, we know that when x = 2, the expression equals 8. This gives us the equation:

a(2)^2 + b(2) + c = 8
4a + 2b + c = 8

Now we have a system of three equations:

1) a + b + c = 0
2) a + b + c = 2
3) 4a + 2b + c = 8

Subtracting equation 1 from equation 2 gives:

0 = 2
This is a contradiction, so there is no solution to the system of equations.

Therefore, the values of a, b, and c cannot be determined.

If ax^2 bx c is divisible by x - 1, then a + b + c = 0.

If ax^2 bx c has a remainder of 2 when divided by x - 1, then a + b = 2.

If ax^2 bx c has a remainder of 8 when divided by x - 2, then 2a + b = 8.

Solving the system of equations:
a + b + c = 0
a + b = 2
2a + b = 8

We get:
a = 3
b = -1
c = 2

Therefore, the values of a, b, and c are 3, -1, and 2, respectively.