Understanding the Problem
To determine the value of "a," we need to analyze the given polynomials:
- Polynomial 1: x^2 + px + q
- Polynomial 2: x^2 + mx + 1
The highest common factor (HCF) is denoted as (x + a).
Finding the HCF
- The HCF of two quadratic polynomials must be of degree 1 or 2.
- Since (x + a) is a linear polynomial, we can assume that it divides both polynomials.
Using Polynomial Division
- For (x + a) to be a factor of both polynomials, substituting x = -a in both equations should yield zero.
1. For the first polynomial:
- Substitute x = -a:
- (-a)^2 + p(-a) + q = 0
- This simplifies to: a^2 - pa + q = 0
2. For the second polynomial:
- Substitute x = -a:
- (-a)^2 + m(-a) + 1 = 0
- This simplifies to: a^2 - ma + 1 = 0
Setting Up Equations
- We now have two equations:
1. a^2 - pa + q = 0
2. a^2 - ma + 1 = 0
- We can equate the two expressions for a^2:
- pa - q = ma - 1
Finding "a"
- Rearranging gives:
- (p - m)a = q - 1
- Therefore, a = (q - 1) / (p - m)
Conclusion
- The value of "a" depends on the coefficients p, q, and m.
- Thus, "a" can be computed directly from the relationship established above, given the values of p, q, and m.

Q-m /p-1