Algebra

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.