`(x-a) (x-b) (a-b) +(x-b) (x-c) (b-c) +(x -c) (x -a) (c-a)` is equal to which of the following?

Options

A. `(a-b) (b-c) (c-a)`
B. `(x-a) (x-b) (x-c)`
C. `-(a-b) (b-c) (c-a)`
D. `-(x-a) (x-b) (x-c)`

Answer

To solve the expression \((x-a)(x-b)(a-b) + (x-b)(x-c)(b-c) + (x-c)(x-a)(c-a)\), we will expand and simplify each term step by step.

### Step 1: Expand Each Term
We will expand each of the three terms in the expression.

1. **First Term: \((x-a)(x-b)(a-b)\)**
- First, expand \((x-a)(x-b)\):
\[ (x-a)(x-b) = x^2 - (a+b)x + ab \]
- Now multiply by \((a-b)\):
\[ (x^2 - (a+b)x + ab)(a-b) = (a-b)x^2 - (a-b)(a+b)x + (a-b)ab \]

2. **Second Term: \((x-b)(x-c)(b-c)\)**
- Expand \((x-b)(x-c)\):
\[ (x-b)(x-c) = x^2 - (b+c)x + bc \]
- Now multiply by \((b-c)\):
\[ (x^2 - (b+c)x + bc)(b-c) = (b-c)x^2 - (b-c)(b+c)x + (b-c)bc \]

3. **Third Term: \((x-c)(x-a)(c-a)\)**
- Expand \((x-c)(x-a)\):
\[ (x-c)(x-a) = x^2 - (c+a)x + ca \]
- Now multiply by \((c-a)\):
\[ (x^2 - (c+a)x + ca)(c-a) = (c-a)x^2 - (c-a)(c+a)x + (c-a)ca \]

### Step 2: Combine All Terms
Now we will combine all the expanded terms:
\[ \text{Total} = \left((a-b)x^2 - (a-b)(a+b)x + (a-b)ab\right) + \left((b-c)x^2 - (b-c)(b+c)x + (b-c)bc\right) + \left((c-a)x^2 - (c-a)(c+a)x + (c-a)ca\right) \]

### Step 3: Factor Out Common Terms
- Combine the \(x^2\) terms:
\[ (a-b + b-c + c-a)x^2 = 0 \quad \text{(since \(a-b + b-c + c-a = 0\))} \]

- Combine the \(x\) terms:
\[ -((a-b)(a+b) + (b-c)(b+c) + (c-a)(c+a))x \]

- Combine the constant terms:
\[ (a-b)ab + (b-c)bc + (c-a)ca \]

### Step 4: Final Expression
Since the coefficient of \(x^2\) is zero, we only need to consider the linear and constant terms. The expression simplifies to:
\[ -((a-b)(a+b) + (b-c)(b+c) + (c-a)(c+a))x + \text{(constant terms)} \]

### Conclusion
After simplification, we find that the expression is equal to:
\[ (a-b)(b-c)(c-a) \]