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, 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 for a, b, and c that satisfies all three conditions.

Remainder Theorem:

When a polynomial f(x) is divided by (x - a), the remainder is f(a).

Step 1: Find a

Since the polynomial is divisible by (x - 1), a = 0.

Step 2: Write the polynomial in terms of b and c

0x^2 + bx + c = bx + c

Step 3: Find b

When divided by (x - 1), the remainder is 2. By the Remainder Theorem, 2 = b(1) + c. So, b = 2 - c.

Step 4: Find c

When divided by (x - 2), the remainder is 8. By the Remainder Theorem, 8 = b(2) + c = 2(2 - c) + c = 4 - 2c + c = 4 - c. So, c = 4.

Step 5: Find b

b = 2 - c = 2 - 4 = -2.

Therefore, the values of a, b, c are:

a = 0
b = -2
c = 4