Question 1209717: Let f(x) be a polynomial. Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 2.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the remainder:
1. **The Remainder Theorem:** When a polynomial f(x) is divided by (x - c), the remainder is f(c). We're given f(0) = 0, f(1) = 1, and f(2) = 2.
2. **Form of the Remainder:** Since we're dividing by a cubic (x(x - 1)(x - 2)), the remainder will be a polynomial of degree at most 2. Let the remainder be R(x) = ax² + bx + c.
3. **Set up equations:** We know the following:
* R(0) = f(0) = 0
* R(1) = f(1) = 1
* R(2) = f(2) = 2
Substitute these values into R(x):
* R(0) = a(0)² + b(0) + c = 0 => c = 0
* R(1) = a(1)² + b(1) + c = 1 => a + b + c = 1
* R(2) = a(2)² + b(2) + c = 2 => 4a + 2b + c = 2
4. **Solve for a and b:** We already know c = 0. Substitute this into the other two equations:
* a + b = 1
* 4a + 2b = 2
Notice that the second equation is just 2 times the first equation. This means the two equations are dependent and there are infinitely many solutions for a and b.
Let's use the first equation to express b in terms of a:
b = 1 - a
Substitute this into the second equation:
4a + 2(1-a) = 2
4a + 2 - 2a = 2
2a = 0
a = 0
Then, b = 1-a = 1-0 = 1.
Therefore, a = 0 and b = 1.
5. **Write the remainder:** Since a = 0, b = 1, and c = 0, the remainder is:
R(x) = 0x² + 1x + 0 = x
Therefore, the remainder when f(x) is divided by x(x - 1)(x - 2) is simply *x*.
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