Cubic Equation: bounds
- Ryan Hirst
I want to answer this question without solving for x: Q:In what range of x do the zeros fall?
Consider the leading term, For some x>0, it overtakes each remaining term, and escapes to . Breaking into three pieces, And . I have three polynomials with zeros I can find. Let be the zero past which is everywhere increasing. Then at , all three of the are positive, increasing. Hence, so is their sum, . A lower bound can be found in the same way. The resulting are Attention to the relative magnitudes of the terms brings tighter bounds. In general, we can break up however we wish, and obtain different bounds. For now, I just want a, b to be finite quantities close enough to true zeros for speedy root-hunting. We can also use information abut the curve (local minimum, maximum, inflection point) to narrow the interval further. __________ Roots of the cubic equation Solution 1 (Iterative): http://www.geogebratube.org/material/show/id/143036 Solution 2: (Algebraic): http://tube.geogebra.org/material/show/id/143424