If for a non - zero `x, 3 x^(2) + 5 x + 3 = 0 ` then the value of ` x^(3) + (1)/( x^(3)) ` is

Options

A. `(10)/(27)`
B. `-((10)/(27))`
C. `(2)/(3)`
D. `-((2)/(3))`

Answer

To solve the equation \(3x^2 + 5x + 3 = 0\) and find the value of \(x^3 + \frac{1}{x^3}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 3x^2 + 5x + 3 = 0 \]

### Step 2: Divide the equation by \(x\)
Since \(x\) is non-zero, we can divide the entire equation by \(x\):
\[ 3x + 5 + \frac{3}{x} = 0 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ 3x + \frac{3}{x} = -5 \]

### Step 4: Factor out 3
We can factor out 3 from the left-hand side:
\[ 3\left(x + \frac{1}{x}\right) = -5 \]

### Step 5: Solve for \(x + \frac{1}{x}\)
Dividing both sides by 3:
\[ x + \frac{1}{x} = -\frac{5}{3} \]

### Step 6: Use the identity for \(x^3 + \frac{1}{x^3}\)
We use the identity:
\[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) \]
Let \(z = x + \frac{1}{x}\). Substituting \(z = -\frac{5}{3}\):
\[ x^3 + \frac{1}{x^3} = \left(-\frac{5}{3}\right)^3 - 3\left(-\frac{5}{3}\right) \]

### Step 7: Calculate \(\left(-\frac{5}{3}\right)^3\)
Calculating the cube:
\[ \left(-\frac{5}{3}\right)^3 = -\frac{125}{27} \]

### Step 8: Calculate \(3\left(-\frac{5}{3}\right)\)
Calculating the second term:
\[ -3\left(-\frac{5}{3}\right) = 5 \]

### Step 9: Combine the results
Now we combine the two results:
\[ x^3 + \frac{1}{x^3} = -\frac{125}{27} + 5 \]
To combine these, we convert 5 to a fraction with a denominator of 27:
\[ 5 = \frac{135}{27} \]
So,
\[ x^3 + \frac{1}{x^3} = -\frac{125}{27} + \frac{135}{27} = \frac{10}{27} \]

### Final Answer
Thus, the value of \(x^3 + \frac{1}{x^3}\) is:
\[ \frac{10}{27} \]

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