If `(x^(3)+3x)/(3x^(2)+1)=(341)/(91)` then the value of x is

Options

A. 11
B. 13
C. 14
D. 16

Answer

To solve the equation \(\frac{x^3 + 3x}{3x^2 + 1} = \frac{341}{91}\), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions:
\[ (x^3 + 3x) \cdot 91 = (3x^2 + 1) \cdot 341 \]

### Step 2: Expand Both Sides
Now, we expand both sides:
\[ 91x^3 + 273x = 1023x^2 + 341 \]

### Step 3: Rearrange the Equation
Next, we rearrange the equation to bring all terms to one side:
\[ 91x^3 - 1023x^2 + 273x - 341 = 0 \]

### Step 4: Factor or Simplify the Polynomial
This polynomial can be complex to factor directly, so we will look for rational roots or use synthetic division. However, we can also apply the Rational Root Theorem or numerical methods to find possible roots.

### Step 5: Test Possible Rational Roots
We can test small integer values for \(x\). Let's start with \(x = 6\):
\[ 91(6^3) - 1023(6^2) + 273(6) - 341 \]
Calculating each term:
- \(6^3 = 216\) so \(91 \cdot 216 = 19656\)
- \(6^2 = 36\) so \(1023 \cdot 36 = 36828\)
- \(273 \cdot 6 = 1638\)

Putting it all together:
\[ 19656 - 36828 + 1638 - 341 = 0 \]
This shows that \(x = 6\) is a root.

### Step 6: Conclusion
Thus, the value of \(x\) is:
\[ \boxed{6} \]