Factors of `(x^(2)+(x)/(6)-(1)/(6))` are

Options

A. `(1)/(6),(2x+1),(3x+1)`
B. `(1)/(6),(2x+1),(3x-1)`
C. `(1)/(6),(2x-1),(3x-1)`
D. `(1)/(6),(2x-1),(3x+1)`

Answer

To find the factors of the expression \( x^2 + \frac{x}{6} - \frac{1}{6} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^2 + \frac{x}{6} - \frac{1}{6} \]
To eliminate the fractions, we will multiply the entire expression by 6 (the least common multiple of the denominators).

### Step 2: Multiply by 6
Multiplying the expression by 6 gives:
\[ 6 \left( x^2 + \frac{x}{6} - \frac{1}{6} \right) = 6x^2 + x - 1 \]

### Step 3: Factor the quadratic expression
Now, we need to factor the quadratic expression \( 6x^2 + x - 1 \). We will look for two numbers that multiply to \( 6 \times (-1) = -6 \) and add to \( 1 \) (the coefficient of x).

The numbers that satisfy this are \( 3 \) and \( -2 \).

### Step 4: Rewrite the middle term
We can rewrite the expression by splitting the middle term:
\[ 6x^2 + 3x - 2x - 1 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (6x^2 + 3x) + (-2x - 1) \]

### Step 6: Factor by grouping
Now we factor out the common factors from each group:
\[ 3x(2x + 1) - 1(2x + 1) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (2x + 1) \):
\[ (3x - 1)(2x + 1) \]

### Step 8: Write the final expression
Now, we can express the original expression with the factor of \( \frac{1}{6} \) that we multiplied out earlier:
\[ \frac{1}{6}(3x - 1)(2x + 1) \]

### Final Answer
Thus, the factors of \( x^2 + \frac{x}{6} - \frac{1}{6} \) are:
\[ \frac{1}{6}(3x - 1)(2x + 1) \]