Algebra

You've presented a very similar problem to the last one, but with a slight change in the value of f(2). Let's work through it.

Understanding the Problem

f(x) is a polynomial.

f(0) = 0

f(1) = 1

f(2) = 2

We want to find the remainder when f(x) is divided by x(x - 1)(x - 2).

Using the Remainder Theorem

As before, we can write:

f(x) = q(x)x(x - 1)(x - 2) + r(x)

where q(x) is the quotient and r(x) is the remainder. Since the divisor is cubic, the remainder is at most quadratic:

r(x) = 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) = 2:

2 = q(2) * 0 + a(2)² + b(2) + c

2 = 4a + 2b + c

Since c = 0, we have 4a + 2b = 2, which simplifies to 2a + b = 1

Solving for a and b

We have the system of equations:

a + b = 1

2a + b = 1

Subtract the first equation from the second equation:

(2a + b) - (a + b) = 1 - 1

a = 0

Substitute a = 0 into a + b = 1:

0 + b = 1

b = 1

The Remainder

We found:

a = 0

b = 1

c = 0

Therefore, the remainder r(x) is:

r(x) = 0x² + 1x + 0 = x

Conclusion

The remainder when f(x) is divided by x(x - 1)(x - 2) is x.