If `P(x)=1/(sqrt(3x+1))[((1+sqrt(3x+1))/5)^(n)-((1-sqrt(3x+1))/5)^(n)]` is a 5th degree polynomial, then value of n is

Options

A. 9
B. 11
C. 23
D. 21

Answer

To find the value of \( n \) such that \( P(x) \) is a 5th degree polynomial, we start with the given expression:

\[ P(x) = \frac{1}{\sqrt{3x + 1}} \left[ \left( \frac{1 + \sqrt{3x + 1}}{5} \right)^n - \left( \frac{1 - \sqrt{3x + 1}}{5} \right)^n \right] \]

### Step 1: Analyze the structure of \( P(x) \)

We can see that \( P(x) \) is a polynomial in \( \sqrt{3x + 1} \) divided by \( \sqrt{3x + 1} \). This suggests that the expression inside the brackets must yield a polynomial of degree 6 in \( \sqrt{3x + 1} \) so that when divided by \( \sqrt{3x + 1} \), the resulting polynomial is of degree 5.

### Step 2: Expand the terms using the Binomial Theorem

Using the Binomial Theorem, we can expand both terms in the brackets:

\[ \left( \frac{1 + \sqrt{3x + 1}}{5} \right)^n = \sum_{k=0}^{n} \binom{n}{k} \left( \frac{1}{5} \right)^{n-k} \left( \sqrt{3x + 1} \right)^k \]

\[ \left( \frac{1 - \sqrt{3x + 1}}{5} \right)^n = \sum_{k=0}^{n} \binom{n}{k} \left( \frac{1}{5} \right)^{n-k} \left( -\sqrt{3x + 1} \right)^k \]

### Step 3: Combine the expansions

When we subtract the two expansions, the even powers of \( \sqrt{3x + 1} \) will cancel out, and we will be left with only the odd powers:

\[ P(x) = \frac{1}{\sqrt{3x + 1}} \left[ 2 \sum_{k \text{ odd}} \binom{n}{k} \left( \frac{1}{5} \right)^{n-k} \left( \sqrt{3x + 1} \right)^k \right] \]

### Step 4: Determine the degree of the polynomial

The highest power of \( \sqrt{3x + 1} \) that remains after the subtraction is \( n \) when \( n \) is odd. Therefore, the degree of the polynomial \( P(x) \) will be \( n - 1 \) (since we divide by \( \sqrt{3x + 1} \)).

### Step 5: Set the degree equal to 5

To ensure \( P(x) \) is a polynomial of degree 5, we need:

\[ n - 1 = 5 \implies n = 6 \]

### Step 6: Verify the conditions

Since we are looking for the highest odd term, we check if \( n \) can be greater than 6. If \( n \) is increased, the degree of \( P(x) \) will also increase. Thus, the maximum value of \( n \) that keeps \( P(x) \) as a 5th degree polynomial is indeed \( n = 6 \).

### Conclusion

Thus, the value of \( n \) such that \( P(x) \) is a 5th degree polynomial is:

\[ \boxed{6} \]