The volume of a cube is given by `V = x^(3) - 9x^(2) + 27x - 27`. The edge of the cube is

Options

A. (x+3)
B. 3(x-3)
C. (x-3)
D. None of these

Answer

To find the edge of the cube given its volume \( V = x^3 - 9x^2 + 27x - 27 \), we can follow these steps:

### Step 1: Recognize the volume formula
The volume \( V \) of a cube can be expressed as \( V = \text{(edge)}^3 \). Therefore, we need to express the given polynomial in the form of a cube.

### Step 2: Factor the polynomial
We need to factor the polynomial \( x^3 - 9x^2 + 27x - 27 \). We can try to express it as a perfect cube.

### Step 3: Identify the structure
We can rewrite the polynomial in a way that resembles the identity for the cube of a binomial:
\[ a^3 - 3a^2b + 3ab^2 - b^3 = (a-b)^3 \]
Here, we will try to express \( x^3 - 9x^2 + 27x - 27 \) in the form \( (x - b)^3 \).

### Step 4: Compare coefficients
To match the polynomial to the form \( (x - 3)^3 \), we can expand \( (x - 3)^3 \):
\[ (x - 3)^3 = x^3 - 3 \cdot 3x^2 + 3 \cdot 3^2x - 3^3 = x^3 - 9x^2 + 27x - 27 \]
This matches our polynomial exactly.

### Step 5: Conclude the edge of the cube
Since we have factored the volume as:
\[ V = (x - 3)^3 \]
This means that the edge of the cube is:
\[ \text{edge} = x - 3 \]

### Final Answer
Thus, the edge of the cube is \( x - 3 \).

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