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# How could you use Descartes' rule to predict the number of complex roots to a polynomial?

Updated: 4/28/2022

Wiki User

12y ago

Descartes' rule of signs will not necessarily tell exact number of complex roots, but will give an idea. The Wikipedia article explains it pretty well, but here is a brief explanation:

It is for single variable polynomials.

• Order the polynomial in descending powers [example: f(x) = axÂ³ + bxÂ² + cx + d]
• Count number of sign changes between consecutive coefficients. This is the maximum possible real positive roots.
• Let function g(x) = f(-x), count number of sign changes, which is maximum number of real negative roots.

Note these are maximums, not the actual numbers. Let p = positive maximum and q = negative maximum. Let m be the order (maximum power of the variable), which is also the total number of roots.

So m - p - q = minimum number of complex roots. Note complex roots always occur in pairs, so number of complex roots will be {0, 2, 4, etc}.

Wiki User

12y ago