Let's solve this problem step-by-step.
Understanding the Problem
We are given:
f(x) is a polynomial.
f(0) = 0
f(1) = 1
f(2) = 4
We want to find the remainder when f(x) is divided by x(x - 1)(x - 2).
Using the Remainder Theorem
When a polynomial f(x) is divided by a divisor d(x), we can express f(x) as:
f(x) = q(x)d(x) + r(x)
where q(x) is the quotient and r(x) is the remainder.
In our case, d(x) = x(x - 1)(x - 2), which is a cubic polynomial. Therefore, the remainder r(x) must be a polynomial of degree at most 2.
Let r(x) = ax² + bx + c.
Then:
f(x) = q(x)x(x - 1)(x - 2) + ax² + bx + c
Applying the Given Information
f(0) = 0:
0 = q(0) * 0 + a(0)² + b(0) + c
c = 0
f(1) = 1:
1 = q(1) * 0 + a(1)² + b(1) + c
1 = a + b + c
Since c = 0, we have a + b = 1
f(2) = 4:
4 = q(2) * 0 + a(2)² + b(2) + c
4 = 4a + 2b + c
Since c = 0, we have 4a + 2b = 4, which simplifies to 2a + b = 2
Solving for a and b
We have the following system of equations:
a + b = 1
2a + b = 2
Subtract the first equation from the second equation:
(2a + b) - (a + b) = 2 - 1
a = 1
Substitute a = 1 into a + b = 1:
1 + b = 1
b = 0
The Remainder
We found:
a = 1
b = 0
c = 0
Therefore, the remainder r(x) is:
r(x) = 1x² + 0x + 0 = x²
Conclusion
The remainder when f(x) is divided by x(x - 1)(x - 2) is x².
