Factors of `1/3 c^2 - 2c - 9` are

Options

A. `(1/3 c + 3 ) (c+3)`
B. `(1/3 c-3) (c-3)`
C. `(1/3 c-3) (c+3)`
D. `(c-1/3)(3c+1)`

Answer

To factor the expression \( \frac{1}{3}c^2 - 2c - 9 \), we can follow these steps:

### Step 1: Factor out the common term
First, we notice that the expression has a common factor of \( \frac{1}{3} \). We can factor this out:

\[ \frac{1}{3}(c^2 - 6c - 27) \]

### Step 2: Rewrite the quadratic expression
Now we need to factor the quadratic expression \( c^2 - 6c - 27 \). We are looking for two numbers that multiply to \(-27\) (the product of the constant term) and add to \(-6\) (the coefficient of the linear term).

### Step 3: Find the factors
The two numbers that satisfy these conditions are \(-9\) and \(3\) because:

\[ -9 \times 3 = -27 \quad \text{and} \quad -9 + 3 = -6 \]

### Step 4: Rewrite the quadratic using the factors
We can now rewrite the quadratic expression using these factors:

\[ c^2 - 6c - 27 = (c - 9)(c + 3) \]

### Step 5: Combine with the common factor
Now we substitute back the common factor we factored out in Step 1:

\[ \frac{1}{3}(c - 9)(c + 3) \]

### Step 6: Final expression
Thus, the final factored form of the expression \( \frac{1}{3}c^2 - 2c - 9 \) is:

\[ \frac{1}{3}(c - 9)(c + 3) \]

### Step 7: Simplifying further (if needed)
We can express this as:

\[ \frac{1}{3}c - 3 \quad \text{and} \quad c + 3 \]

So the factors of the expression are:

\[ \left(\frac{1}{3}c - 3\right)(c + 3) \]

### Final Answer
The factors of \( \frac{1}{3}c^2 - 2c - 9 \) are \( \left(\frac{1}{3}c - 3\right)(c + 3) \).

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