Question

Determine the consecutive values of x between which each real zero is located. -11x^4-3x^3-10x^2+9x+18

Answer 1

\[-11x^4-3x^3-10x^2+9x+18\]

I would start out by evaluating this at x = 0. Note whether the result is positive or negative (or zero, but it isn't in this case). Next, evaluate at x = 1 and x = -1. If the value is positive at one point and negative at the next (or vice versa), there must have been a zero in between those two points. Fortunately, as you move away from the y-axis, the effect of the \(x^4\) term drives the value of the function away from the x-axis, never to return, so you don't have to typically try very many points to find all of the intervals where there is a root.

Now, a different question is how many roots should we expect to see. Because this polynomial has \(x^4\) as the highest power, we have 4 roots. Some number of pairs of them may be complex conjugate roots, of the form \(a\pm bi\) where \(i=\sqrt{-1}\). If you know Descartes' Rule of Signs, you can analyze the possible combinations of positive real zeros, negative real zeros, and pairs of complex zeros. Because the polynomial has only real coefficients, any complex roots must come in the aforementioned conjugate pairs. Description of how to use Descartes' Rule of Signs available on request.