IF the equations `x^(3) + 5x^(2) + px + q = 0 and x^(3) + 7x^(2) + px + r = 0` have two roots in common, then the product of two non-common roots of two equations, is

Options

A. 35
B. -35
C. 35 + p - q
D. 35 + p + q - r

Explanation

To solve the problem, we need to analyze the two cubic equations given and find the product of the two non-common roots. Let's denote the roots of the first equation as \( \alpha, \beta, x_1 \) and the roots of the second equation as \( \alpha, \beta, x_2 \).

### Step-by-Step Solution:

1. **Identify the equations and their roots:**
- The first equation is \( x^3 + 5x^2 + px + q = 0 \) with roots \( \alpha, \beta, x_1 \).
- The second equation is \( x^3 + 7x^2 + px + r = 0 \) with roots \( \alpha, \beta, x_2 \).

2. **Use Vieta's formulas to express the sums of the roots:**
- For the first equation:
\[ \alpha + \beta + x_1 = -5 \]
- For the second equation:
\[ \alpha + \beta + x_2 = -7 \]

3. **Express the product of the roots:**
- For the first equation:
\[ \alpha\beta + \beta x_1 + \alpha x_1 = p \]
- For the second equation:
\[ \alpha\beta + \beta x_2 + \alpha x_2 = p \]

4. **Equate the two equations for \( p \):**
Since both equations equal \( p \), we can set them equal to each other:
\[ \alpha\beta + x_1(\alpha + \beta) = \alpha\beta + x_2(\alpha + \beta) \]

5. **Simplify the equation:**
- Cancel \( \alpha\beta \) from both sides:
\[ x_1(\alpha + \beta) = x_2(\alpha + \beta) \]
- If \( \alpha + \beta \neq 0 \), we can divide both sides by \( \alpha + \beta \):
\[ x_1 = x_2 \]
- However, since \( x_1 \) and \( x_2 \) are roots of different equations, they cannot be equal. Thus, we conclude:
\[ \alpha + \beta = 0 \]

6. **Substitute \( \alpha + \beta = 0 \) back into the equations:**
- From \( \alpha + \beta + x_1 = -5 \):
\[ 0 + x_1 = -5 \implies x_1 = -5 \]
- From \( \alpha + \beta + x_2 = -7 \):
\[ 0 + x_2 = -7 \implies x_2 = -7 \]

7. **Calculate the product of the non-common roots:**
- The non-common roots are \( x_1 \) and \( x_2 \):
\[ x_1 \cdot x_2 = (-5) \cdot (-7) = 35 \]

### Final Answer:
The product of the two non-common roots of the two equations is \( 35 \).