The number of integral roots of the equation `x ^(8) -24x ^(7) -18x ^(5) +39x ^(2) +1155=0` is:

Options

A. 0
B. 2
C. 4
D. 6

Answer

To find the number of integral roots of the equation \( x^8 - 24x^7 - 18x^5 + 39x^2 + 1155 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^8 - 24x^7 - 18x^5 + 39x^2 + 1155 = 0 \]
This can be rewritten as:
\[ x^8 - 24x^7 - 18x^5 + 39x^2 = -1155 \]

### Step 2: Factor out \( x^2 \)
We can factor out \( x^2 \) from the left-hand side:
\[ x^2 (x^6 - 24x^5 - 18x^3 + 39) = -1155 \]

### Step 3: Analyze the factor \( x^2 \)
The term \( x^2 \) implies that if \( x = 0 \), then the left-hand side becomes \( 0 \), which does not satisfy the equation since \( -1155 \neq 0 \). Therefore, \( x = 0 \) is not a root.

### Step 4: Check for other integral roots
We will check for possible integral roots using the Rational Root Theorem, which suggests that any rational root (in this case, integral roots) must be a factor of the constant term \( 1155 \).

### Step 5: Find the prime factorization of \( 1155 \)
The prime factorization of \( 1155 \) is:
\[ 1155 = 3 \times 5 \times 7 \times 11 \]
The integral factors of \( 1155 \) are \( \pm 1, \pm 3, \pm 5, \pm 7, \pm 11, \pm 15, \pm 21, \pm 33, \pm 35, \pm 55, \pm 105, \pm 231, \pm 385, \pm 1155 \).

### Step 6: Test the integral factors
We will test these factors in the original polynomial to see if any yield \( 0 \).

1. **Testing \( x = 1 \)**:
\[ 1^8 - 24 \cdot 1^7 - 18 \cdot 1^5 + 39 \cdot 1^2 + 1155 = 1 - 24 - 18 + 39 + 1155 = 1153 \quad (\text{not a root}) \]

2. **Testing \( x = -1 \)**:
\[ (-1)^8 - 24 \cdot (-1)^7 - 18 \cdot (-1)^5 + 39 \cdot (-1)^2 + 1155 = 1 + 24 + 18 + 39 + 1155 = 1237 \quad (\text{not a root}) \]

3. **Testing \( x = 3 \)**:
\[ 3^8 - 24 \cdot 3^7 - 18 \cdot 3^5 + 39 \cdot 3^2 + 1155 = 6561 - 24 \cdot 2187 - 18 \cdot 243 + 39 \cdot 9 + 1155 \]
This results in a large negative number, so not a root.

4. **Testing \( x = -3 \)**:
\[ (-3)^8 - 24 \cdot (-3)^7 - 18 \cdot (-3)^5 + 39 \cdot (-3)^2 + 1155 \]
This also results in a large positive number, so not a root.

5. **Continuing this process for other factors**:
After testing all integral factors, none yield \( 0 \).

### Conclusion
After testing all possible integral roots, we find that there are no integral roots for the equation \( x^8 - 24x^7 - 18x^5 + 39x^2 + 1155 = 0 \).

Thus, the number of integral roots is:
\[ \boxed{0} \]