Let `alpha,beta,gamma` be the roots of `(x-a) (x-b) (x-c) = d, d != 0`, then the roots of the equation `(x-alpha)(x-beta)(x-gamma) + d =0` are `:`

Options

A. a,b,d
B. b,c,d
C. a,b,c
D. `a+d,b+d,c+d`

Explanation

To solve the problem, we need to find the roots of the equation \((x - \alpha)(x - \beta)(x - \gamma) + d = 0\), given that \(\alpha, \beta, \gamma\) are the roots of the equation \((x - a)(x - b)(x - c) = d\).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ (x - a)(x - b)(x - c) = d \]
This equation can be rewritten in standard polynomial form.

2. **Expand the left-hand side**:
\[ (x - a)(x - b) = x^2 - (a + b)x + ab \]
Now multiply this by \((x - c)\):
\[ (x^2 - (a + b)x + ab)(x - c) = x^3 - (a + b + c)x^2 + (ab + ac + bc)x - abc \]
So we have:
\[ x^3 - (a + b + c)x^2 + (ab + ac + bc)x - abc = d \]

3. **Rearranging the equation**:
\[ x^3 - (a + b + c)x^2 + (ab + ac + bc)x - (abc + d) = 0 \]
This polynomial has roots \(\alpha, \beta, \gamma\).

4. **Now consider the new equation**:
\[ (x - \alpha)(x - \beta)(x - \gamma) + d = 0 \]
Expanding this gives:
\[ x^3 - (\alpha + \beta + \gamma)x^2 + (\alpha\beta + \beta\gamma + \gamma\alpha)x - \alpha\beta\gamma + d = 0 \]

5. **Substituting the values of the sums and products of roots**:
From the previous polynomial, we know:
- \(\alpha + \beta + \gamma = a + b + c\)
- \(\alpha\beta + \beta\gamma + \gamma\alpha = ab + ac + bc\)
- \(\alpha\beta\gamma = abc + d\)

Substituting these into the new equation:
\[ x^3 - (a + b + c)x^2 + (ab + ac + bc)x - (abc + d) + d = 0 \]
Simplifying gives:
\[ x^3 - (a + b + c)x^2 + (ab + ac + bc)x - abc = 0 \]

6. **Recognizing the equation**:
The equation we have now:
\[ x^3 - (a + b + c)x^2 + (ab + ac + bc)x - abc = 0 \]
is the same as the original equation \((x - a)(x - b)(x - c) = 0\).

7. **Conclusion**:
Therefore, the roots of the equation \((x - \alpha)(x - \beta)(x - \gamma) + d = 0\) are:
\[ x = a, b, c \]

### Final Answer:
The roots of the equation \((x - \alpha)(x - \beta)(x - \gamma) + d = 0\) are \(a, b, c\).

---