MATH 110: Materials - Descartes' Rule of Signs

COLLEGE ALGEBRA MATH 110		Southwestern Adventist University
Distance Education	Lawrence E. Turner, Jr., Ph.D.

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Descartes' Rule of Signs

Descartes' Rule of Signs is a relatively easy to apply test which gives information about the existence of real zeros of a polynomial function. It counts the changes in the signs of the coefficients of the terms and gives the possible numbers of positive and negative real zeros (and possible numbers of complex zeros).

Given a polynomial of degree n with real coefficients:

P_n(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + ... + a₁x + a₀, both a_n and a₀ not zero.

Note: for the following process the polynomial must be written "in order" with decreasing powers of x.

Note: if a₀ = 0, then there are one or more common factors of x which can be factored out until there is indeed a constant term. Each common factor of x, of course, corresponds to a zero of x = 0. The resulting polynomial is in the "desired" form with degree less than n.

Positive Real Zeros

The number of positive real zeros is equal to the number of sign changes of P_n(x), moving from the xⁿ term to the constant term (left to right as we normally write a polynomial), or the actual number of positive real zeros differs from the number of sign changes by an even number.

Negative Real Zeros

The number of negative real zeros is equal to the number of sign changes of P_n(−x), moving from the xⁿ term to the constant term (left to right as we normally write a polynomial), or it differs from the number of sign changes by an even number.

Note:

the terms must be written "in order," and we ignore any missing terms; that is, coefficients of 0.
to get all the possible number of real zeros, use the number of sign changes and count down to 0 or 1 by twos.
the polynomial needs to have a constant term, this means we have previously factored out any common factors of x.
the easy way to generate P_n(−x) is to change the sign of each of terms containing an odd power of x.

As an example:

   f(x) = 3x⁵ + 2x⁴ − 5x³ + 4x² + 6x + 7

has signs:   + +_−_+ + +   and moving left-to right, there are 2 changes (indicated by _ ).

Therefore, there are 2 or 0 positive real zeros.

   and

   f(−x) = −3x⁵ + 2x⁴ + 5x³ + 4x² − 6x + 7

has signs:   −_+ + +_−_+   and moving left-to right, there are 3 changes.

Therefore, there are 3 or 1 negative real zeros.

These numbers can be combined in a table to display all the possibilities of the existence of real and complex zeros. Recall, the total number of zeros is n, and all complex zeros occur in pairs.

+ real	− real	complex	total

2	3	0	5
2	1	2	5
0	3	2	5
0	1	4	5

For each of the possible numbers of positive real zeros, we use all the possible numbers for the negative real zeros. Thus the number of rows is equal to the possible number of positive real zeros times the possible number of negative real zeros. For the example above we have two possible values for the number of positive real zeros (2 or 0) and two possible values for the number of negative real zeros (3 or 1) which result in the four rows of the table. The number of complex zeros are determined simply to make the total equal n—the number of complex zeros should always be an even number!

The actual set of zeros corresponds to one of the rows in the table.

For the example above, we see that there may be both positive real zeros as well as negative ones. For other situations, the application of Descarte's Rule of Signs may show that there are no positive real zeros (or no negative real zeros). The usefulness of the Descartes' Rule of Signs is that this procedure is relatively easy to do and may with little effort may eliminate possible zeros and save work!