The equation
`(a(x-b)(x-c))/((a-b)(a-c)) + (b(x-c)(x-a))/((b-c)(b-a))+ (c (x-a) (x-b))/((c-a)(c-b))= x ` is satisfied by

Options

A. No value of x .
B. Exactly two values of x.
C. Exactly three values of x.
D. All values of x

Answer

To solve the equation

\[ \frac{a(x-b)(x-c)}{(a-b)(a-c)} + \frac{b(x-c)(x-a)}{(b-c)(b-a)} + \frac{c(x-a)(x-b)}{(c-a)(c-b)} = x, \]

we will analyze the left-hand side and simplify it step by step.

### Step 1: Define the function
Let

\[ f(x) = \frac{a(x-b)(x-c)}{(a-b)(a-c)} + \frac{b(x-c)(x-a)}{(b-c)(b-a)} + \frac{c(x-a)(x-b)}{(c-a)(c-b)}. \]

### Step 2: Analyze the degree of the polynomial
Notice that each term in \(f(x)\) is a quadratic polynomial in \(x\). Therefore, \(f(x)\) is a polynomial of degree 2.

### Step 3: Set up the equation
We need to solve

\[ f(x) = x. \]

This can be rewritten as

\[ f(x) - x = 0. \]

### Step 4: Determine the form of \(f(x) - x\)
Since \(f(x)\) is a quadratic polynomial, we can express it in the form:

\[ f(x) - x = Ax^2 + Bx + C, \]

where \(A\), \(B\), and \(C\) are constants that depend on \(a\), \(b\), and \(c\).

### Step 5: Find the roots
The equation \(Ax^2 + (B-1)x + C = 0\) is a quadratic equation. The number of solutions (roots) depends on the discriminant \(D\):

\[ D = (B-1)^2 - 4AC. \]

### Step 6: Analyze the roots
1. If \(D > 0\), there are two distinct real roots.
2. If \(D = 0\), there is one real root (a repeated root).
3. If \(D < 0\), there are no real roots.

### Step 7: Special cases
Now, we can check specific values of \(x\):
- If we substitute \(x = a\), \(x = b\), and \(x = c\) into \(f(x)\), we find that \(f(a) = a\), \(f(b) = b\), and \(f(c) = c\). This implies that \(x = a\), \(x = b\), and \(x = c\) are roots of the equation \(f(x) - x = 0\).

### Step 8: Conclusion
Since \(f(x)\) is a quadratic polynomial and we have found three distinct values \(a\), \(b\), and \(c\) that satisfy the equation, we conclude that the equation is satisfied for all values of \(x\).

Thus, the final answer is that the equation is satisfied for all values of \(x\).