To determine the condition under which one of the points A, B, or C bisects the distance between the other two, we need to delve into the properties of the roots of the polynomial equation given. The roots of the equation \( ax^3 + 3bx^2 + 3cx + d = 0 \) represent the distances of points A, B, and C from a fixed origin O on a straight line. 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 be positioned exactly halfway between them. Mathematically, if we assume that point B bisects the distance between points A and C, we can express this relationship as:
- If B bisects A and C: \( r_2 = \frac{r_1 + r_3}{2} \)
Setting Up the Equation
From the bisection condition, we can rearrange the equation:
- Multiply both sides by 2: \( 2r_2 = r_1 + r_3 \)
This equation indicates that the sum of the distances of points A and C from the origin must equal twice the distance of point B from the origin.
Using Vieta's Formulas
Now, let's relate this to the coefficients of the polynomial. According to Vieta's formulas, for a cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), the relationships between the roots and coefficients are as follows:
- Sum of the roots: \( r_1 + r_2 + r_3 = -\frac{3b}{a} \)
- Sum of the products of the roots taken two at a time: \( r_1r_2 + r_2r_3 + r_3r_1 = \frac{3c}{a} \)
- Product of the roots: \( r_1r_2r_3 = -\frac{d}{a} \)
Deriving the Condition
Substituting the bisection condition \( r_1 + r_3 = 2r_2 \) into the sum of the roots gives us:
- From Vieta's: \( r_1 + r_2 + r_3 = -\frac{3b}{a} \)
- Substituting \( r_1 + r_3 = 2r_2 \): \( 2r_2 + r_2 = -\frac{3b}{a} \)
- This simplifies to \( 3r_2 = -\frac{3b}{a} \)
- Thus, \( r_2 = -\frac{b}{a} \)
Now, we can express the condition for bisection in terms of the coefficients of the polynomial. The point that bisects the distance between the other two must be the mean of the roots, which is \( -\frac{b}{a} \).
Conclusion
In summary, for one of the points A, B, or C to bisect the distance between the other two, the condition is that the distance of the bisecting point from the origin must equal \( -\frac{b}{a} \). This relationship ties together the geometric interpretation of the roots with the algebraic properties of the polynomial, providing a clear condition for bisection.
