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: Rearrange the Equation
We start by rearranging the equation to isolate the constant term on one side:
\[ x^8 - 24x^7 - 18x^5 + 39x^2 = -1155 \]

### Step 2: Factor Out \( x^2 \)
Next, 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 Right-Hand Side
Since \( x^2 \) is always non-negative, the left-hand side must be non-positive for the equation to hold true. Therefore, we need to analyze the right-hand side, which is \(-1155\).

### Step 4: Find the Factors of 1155
To find potential integral roots, we need to factor \( 1155 \):
\[ 1155 = 3 \times 5 \times 7 \times 11 \]
The integral factors of \( 1155 \) are \( \pm 1, \pm 3, \pm 5, \pm 7, \pm 11, \pm 15, \pm 21, \pm 33, \pm 35, \pm 55, \pm 77, \pm 105, \pm 165, \pm 231, \pm 385, \pm 1155 \).

### Step 5: Test Possible Integral Roots
We will test the integral factors to see if they satisfy the equation.

1. **For \( x = 0 \)**:
\[ 0^2(0^6 - 24 \cdot 0^5 - 18 \cdot 0^3 + 39) = 0 \quad \text{(not equal to -1155)} \]

2. **For \( x = 1 \)**:
\[ 1^2(1 - 24 - 18 + 39) = 1^2(-2) = -2 \quad \text{(not equal to -1155)} \]

3. **For \( x = -1 \)**:
\[ (-1)^2(1 + 24 + 18 + 39) = 1 \cdot 82 = 82 \quad \text{(not equal to -1155)} \]

4. **For \( x = 2 \)**:
\[ 2^2(64 - 768 - 144 + 39) = 4(-809) = -3236 \quad \text{(not equal to -1155)} \]

5. **For \( x = 3 \)**:
\[ 3^2(729 - 1944 - 486 + 39) = 9(-1662) = -14958 \quad \text{(not equal to -1155)} \]

6. **For \( x = 5 \)**:
\[ 5^2(15625 - 30000 - 2250 + 39) = 25(-16686) = -417150 \quad \text{(not equal to -1155)} \]

7. **For \( x = 7 \)**:
\[ 7^2(117649 - 24 \cdot 7^5 - 18 \cdot 7^3 + 39) \quad \text{(not equal to -1155)} \]

8. **For \( x = 11 \)**:
\[ 11^2(11^6 - 24 \cdot 11^5 - 18 \cdot 11^3 + 39) \quad \text{(not equal to -1155)} \]

### Step 6: Conclusion
After testing all integral factors, we find that none of them satisfy the equation. Therefore, the number of integral roots of the equation is:
\[ \text{Number of integral roots} = 0 \]