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 can follow these steps:

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

### 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: Apply the Rational Root Theorem
To find potential rational roots, we can use the Rational Root Theorem. We will test possible rational roots such as \(x = 1\), \(x = -1\), \(x = 2\), etc.

### Step 5: Testing Rational Roots
Let's test \(x = 11\):
\[ 91(11)^3 - 1023(11)^2 + 273(11) - 341 \]
Calculating each term:
- \(91(11)^3 = 91 \cdot 1331 = 121201\)
- \(1023(11)^2 = 1023 \cdot 121 = 123783\)
- \(273(11) = 3003\)

Putting it all together:
\[ 121201 - 123783 + 3003 - 341 = 0 \]
This shows that \(x = 11\) is indeed a root.

### Step 6: Conclusion
Thus, the value of \(x\) is:
\[ \boxed{11} \]

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