Cubic Equation: bounds

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

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