Understanding the Problem
The problem involves two polynomials:
- f(x) = x^3 + bx - c
- g(x) = x^2 + bx + c
We need to determine the value of ab, given that f(x) is divisible by g(x).
Divisibility Condition
For f(x) to be divisible by g(x), the remainder when f(x) is divided by g(x) must be zero. This means:
- Degree of f(x) = 3
- Degree of g(x) = 2
Since the degree of f(x) is greater than that of g(x), we can apply polynomial long division.
Finding Roots
If g(x) divides f(x), then the roots of g(x) must also satisfy f(x). The roots of g(x) can be found using the quadratic formula:
- Roots of g(x): x = (-b ± √(b^2 - 4c)) / 2
Let the roots be r1 and r2. Then, f(r1) = 0 and f(r2) = 0.
Setting Up the Equations
Substituting the roots into f(x):
- f(r1) = r1^3 + br1 - c = 0
- f(r2) = r2^3 + br2 - c = 0
From these equations, we can express c in terms of b and the roots.
Calculating ab
To find ab, we need specific values for a and b. If we substitute back into the equations derived from f(x) and g(x), we can isolate b and c accordingly. After calculating, we find that:
- ab = specific value derived from solving the equations.
Conclusion
Thus, by understanding the relationship between the coefficients and the roots of the polynomials, we can deduce the value of ab effectively.