To determine the condition under which one of the points A, B, or C bisects the distance between the other two, we need to analyze the roots of the cubic equation given: \( ax^3 + 3bx^2 + 3cx + d = 0 \). The roots of this equation represent the distances of points A, B, and C from a fixed origin O. Let's denote these roots as \( r_1, r_2, \) and \( r_3 \), corresponding to points A, B, and C, respectively.
Understanding the Concept of Bisection
For one point to bisect the distance between the other two, it must satisfy a specific relationship. If we assume that point B bisects the distance between points A and C, then the following condition must hold:
- Point B is the midpoint of points A and C.
This can be mathematically expressed as:
r_2 = (r_1 + r_3) / 2
Deriving the Condition
To derive the condition from the roots of the cubic equation, we can use Vieta's formulas, which relate the coefficients of the polynomial to sums and products of its roots. For the cubic equation \( ax^3 + 3bx^2 + 3cx + d = 0 \), Vieta's gives us:
- The sum of the roots: \( r_1 + r_2 + r_3 = -\frac{3b}{a} \)
- The sum of the products of the roots taken two at a time: \( r_1r_2 + r_2r_3 + r_3r_1 = \frac{3c}{a} \)
- The product of the roots: \( r_1r_2r_3 = -\frac{d}{a} \)
Now, substituting the bisection condition \( r_2 = (r_1 + r_3) / 2 \) into the sum of the roots, we get:
r_1 + \frac{r_1 + r_3}{2} + r_3 = -\frac{3b}{a}
Multiplying through by 2 to eliminate the fraction gives:
2r_1 + r_1 + r_3 + 2r_3 = -\frac{6b}{a}
3r_1 + 3r_3 = -\frac{6b}{a}
From this, we can simplify to:
r_1 + r_3 = -\frac{2b}{a}
Final Condition for Bisection
Now, substituting \( r_1 + r_3 \) back into the bisection condition, we find:
r_2 = \frac{-\frac{2b}{a}}{2} = -\frac{b}{a}
Thus, the condition for point B to bisect the distance between points A and C is:
r_2 = -\frac{b}{a}
Conclusion
In summary, for one of the points A, B, or C to bisect the distance between the other two, we derived that the middle point must equal \(-\frac{b}{a}\) based on the roots of the cubic equation. This relationship can be extended similarly if we consider point A or point C as the bisector, leading to analogous conditions. Understanding these relationships helps in visualizing how points relate to each other on a number line and can be crucial in various applications in geometry and algebra.
