The volume of a cuboid is`x^(3) - 7x + 6`, then the longest side of cuboid is

Options

A. x + 3
B. x-1
C. x-2
D. None of the above

Answer

To find the longest side of the cuboid given its volume as \( V = x^3 - 7x + 6 \), we will follow these steps:

### Step 1: Factor the polynomial
The volume of the cuboid is given by the polynomial \( V = x^3 - 7x + 6 \). We need to factor this polynomial to express it as a product of three linear factors.

### Step 2: Find a root of the polynomial
We can start by checking for rational roots using the Rational Root Theorem. Let's test \( x = 1 \):

\[ V(1) = 1^3 - 7(1) + 6 = 1 - 7 + 6 = 0 \]

Since \( V(1) = 0 \), \( x - 1 \) is a factor of the polynomial.

### Step 3: Perform polynomial long division
Now, we will divide \( x^3 - 7x + 6 \) by \( x - 1 \):

1. Divide the leading term: \( x^3 \div x = x^2 \).
2. Multiply \( x^2 \) by \( x - 1 \): \( x^2(x - 1) = x^3 - x^2 \).
3. Subtract:
\[ (x^3 - 7x + 6) - (x^3 - x^2) = x^2 - 7x + 6 \]
4. Repeat the process with \( x^2 - 7x + 6 \):
- Divide the leading term: \( x^2 \div x = x \).
- Multiply: \( x(x - 1) = x^2 - x \).
- Subtract:
\[ (x^2 - 7x + 6) - (x^2 - x) = -6x + 6 \]
5. Finally, divide \( -6x + 6 \) by \( x - 1 \):
- Divide: \( -6x \div x = -6 \).
- Multiply: \( -6(x - 1) = -6x + 6 \).
- Subtract:
\[ (-6x + 6) - (-6x + 6) = 0 \]

So, we have \( x^3 - 7x + 6 = (x - 1)(x^2 + x - 6) \).

### Step 4: Factor the quadratic
Next, we factor \( x^2 + x - 6 \):
We need two numbers that multiply to \(-6\) and add to \(1\). These numbers are \(3\) and \(-2\).

Thus, we can factor it as:
\[ x^2 + x - 6 = (x + 3)(x - 2) \]

### Step 5: Write the complete factorization
Now we can write the complete factorization of the volume:
\[ x^3 - 7x + 6 = (x - 1)(x + 3)(x - 2) \]

### Step 6: Identify the longest side
The factors represent the dimensions of the cuboid:
- Length: \( x - 1 \)
- Width: \( x + 3 \)
- Height: \( x - 2 \)

To find the longest side, we compare the three expressions:
- \( x - 1 \)
- \( x + 3 \)
- \( x - 2 \)

Clearly, \( x + 3 \) is the largest since it has the highest constant added to \( x \).

### Final Answer
Thus, the longest side of the cuboid is:
\[ \boxed{x + 3} \]