Descartes’s Rule of Signs
Context
When do I use it?
What does it tell me?
What else do I get?
When I have a polynomial 𝑃(𝑥) with real number coefficients (fractions, irrationals ok!).
The possible counts of the positive real number zeros of 𝑃(𝑥), and
The possible counts of the negative real number zeros of 𝑃(𝑥).
The possible counts of the nonreal complex zeros of 𝑃(𝑥).
Background knowledge
(−𝑥)𝑒𝑣𝑒𝑛 𝑝𝑜𝑤𝑒𝑟
→ 𝑥𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑣𝑒𝑛 𝑝𝑜𝑤𝑒𝑟 (stays the same) but (−𝑥)𝑜𝑑𝑑 𝑝𝑜𝑤𝑒𝑟 → −𝑥𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑜𝑑𝑑 𝑝𝑜𝑤𝑒𝑟 (the sign changes)
So if you have 𝑃(𝑥) = some polynomial, 𝑃(−𝑥) = the same polynomial with the signs changed on any terms of
odd degree. The terms of even degree are unchanged.
How to use Descartes’s Rule of Signs
What to do
Count the sign
changes of 𝑷(𝒙)
Form 𝑷(−𝒙) and
count its sign
changes.
Think about
possibilities
What does it mean?
How many positive real
number zeros could 𝑃(𝑥)
possibly have?
How many negative real
number zeros could 𝑃(𝑥)
possibly have?
There could be nonreal
complex zeros, too.
Example 𝑷(𝒙) = 𝒙𝟔 + 𝒙𝟓 − 𝟐𝒙𝟒 − 𝟓𝒙𝟑 + 𝒙𝟐 − 𝟕𝒙 + 𝟑
It has 4 sign changes.
So 𝑃(𝑥) has either 4 or 2 or 0 positive real zeros.
𝑷(−𝒙) = 𝒙𝟔 − 𝒙𝟓 − 𝟐𝒙𝟒 + 𝟓𝒙𝟑 + 𝒙𝟐 + 𝟕𝒙 + 𝟑
It has 2 sign changes.
So 𝑃(𝑥) has either 2 or 0 negative real zeros.
The degree of 𝑃(𝑥) is 6 so 𝑃(𝑥) has 6 zeros.
𝑃(𝑥) could have 0 or 2 or 4 or 6 nonreal complex zeros.
(It turns out that this example has 2 positive real zeros, both irrational, 0 negative real zeros, and 4 nonreal complex zeros.)
071_DescartesRuleOfSigns.docx
12/23/2013 2:10 PM D.R.S.