A polynomial of 6th degree `f(x)` satisfies `f(x)=f(2-x), AA x in R`, if `f(x)=0` has 4 distinct and 2 equal roots, then sum of the roots of `f(x)=0` is

Options

A. (a) 4
B. (b) 5
C. (c) 6
D. (d) 7

Answer

To solve the problem, we need to find the sum of the roots of the polynomial \( f(x) \) given the conditions in the question.

### Step-by-step Solution:

1. **Understanding the Symmetry**:
We know that \( f(x) = f(2 - x) \) for all \( x \in \mathbb{R} \). This indicates that the polynomial is symmetric about the line \( x = 1 \). This means that if \( r \) is a root, then \( 2 - r \) is also a root.

**Hint**: Recognize that symmetry about \( x = 1 \) implies that roots can be paired.

2. **Identifying the Roots**:
The polynomial \( f(x) \) is of degree 6, and it has 4 distinct roots and 2 equal roots. Since the polynomial is symmetric about \( x = 1 \), the two equal roots must be at the point of symmetry, which is \( x = 1 \).

**Hint**: The equal roots must be located at the center of the symmetry.

3. **Setting Up the Roots**:
Let the distinct roots be \( a, b, c, d \). Since the polynomial has two equal roots at \( x = 1 \), we can express the roots as:
- Roots: \( 1, 1, a, b, c, d \)

**Hint**: Remember that the two equal roots contribute twice to the sum.

4. **Using Symmetry for Distinct Roots**:
Because of the symmetry, if \( a \) is a root, then \( 2 - a \) must also be a root. Therefore, we can pair the distinct roots:
- Let \( a = 1 - p \) and \( b = 1 + p \) (for some \( p \)).
- Let \( c = 1 - q \) and \( d = 1 + q \) (for some \( q \)).

**Hint**: Use variables to express the distinct roots in terms of their distance from the center.

5. **Calculating the Sum of the Roots**:
The sum of the roots can now be calculated as follows:
\[ \text{Sum of roots} = 1 + 1 + (1 - p) + (1 + p) + (1 - q) + (1 + q) \]
Simplifying this:
\[ = 2 + 1 + 1 + 1 + 1 = 6 \]

**Hint**: When summing, notice that the \( p \) and \( q \) terms cancel out.

6. **Final Answer**:
Therefore, the sum of the roots of \( f(x) = 0 \) is \( 6 \).

### Conclusion:
The sum of the roots of the polynomial \( f(x) = 0 \) is \( \boxed{6} \).