Step 1: Understanding the Concept:
When two cubic equations share two roots, their difference leads to a simpler equation (quadratic or linear) that must also be satisfied by those common roots. We use Vieta's formulas to relate the roots of each individual cubic.
Step 2: Key Formula or Approach:
1. Let the common roots be \( \alpha, \beta \).
2. Use sum of roots: \( \sum x_i = -a_{n-1} \).
3. Subtract the two equations to find a condition on the common roots.
Step 3: Detailed Explanation:
Let the common roots be \( \alpha \) and \( \beta \).
For the first equation \( x^3 + 5x^2 + px + q = 0 \):
Roots are \( \alpha, \beta, x_1 \). Sum of roots: \( \alpha + \beta + x_1 = -5 \quad \dots(1) \).
For the second equation \( x^3 + 7x^2 + px + r = 0 \):
Roots are \( \alpha, \beta, x_2 \). Sum of roots: \( \alpha + \beta + x_2 = -7 \quad \dots(2) \).
Subtracting the two equations:
\[ (x^3 + 7x^2 + px + r) - (x^3 + 5x^2 + px + q) = 0 \]
\[ 2x^2 + (r - q) = 0 \implies 2x^2 = q - r \]
Since \( \alpha \) and \( \beta \) are common roots, they must satisfy this resulting quadratic.
\[ \alpha^2 = \frac{q - r}{2} \text{ and } \beta^2 = \frac{q - r}{2} \]
This implies \( \beta = -\alpha \).
Substituting \( \beta = -\alpha \) into the sum of roots:
\[ \alpha + (-\alpha) + x_1 = -5 \implies x_1 = -5 \]
\[ \alpha + (-\alpha) + x_2 = -7 \implies x_2 = -7 \]
The third roots are -5 and -7.
The Greatest Common Divisor (GCD) of -5 and -7 is:
\[ \text{GCD}(5, 7) = 1 \]
Step 4: Final Answer:
The third roots are -5 and -7, and their GCD is 1.