De Carte's Rule of Signs

Pronunciation: /deɪˈkɑrtz rul ʌv saɪnz/ ?

De Carte's rule of signs is a rule for determining the maximum number of positive zeros of a polynomial.[1] For any polynomial, the maximum number of positive zeros is equal to the number of sign changes of the polynomial going from the term with the highest degree to the term with the least degree. If there are a maximum of n positive roots, the exact number of roots is one of n, n-2, n-4, .... Terms with a coefficient of 0 are ignored.

The maximum number of negative zeros of the polynomial is equal to the number of sign changes for f(-x). This changes sign of the terms of odd degree. Each time a non-zero term changes sign from the previous non-zero term, it counts as a sign change. Any roots that are neither negative or positive are complex roots.

Examples

TermsSign
Changes
Positive
Roots
f(-x)Sign
Changes
Negative
Roots
Complex
Roots
3x2 + 2003x2 + 2002
x - 211-x-2000
3x2 + 2x + 7003x2-2x+722 or 00 or 2
-x3 - x2 + 511x3-x2+522 or 00 or 2
2x4 - 3x2 + 222 or 02x4-3x2+222 or 00, 2, or 4
4x6-2x5-3x4-x3+5x2+4x-1 = 033 or 14x6+2x5-3x4+x3+5x2-4x-1 = 033 or 12, 4, or 6
Table 1

References

  1. Ron Larson, Robert P. Hostetler. Precalculus, 7th edition, pg 176. Brooks Cole, January 26, 2006. (Accessed: 2010-01-22).

Printed Resources

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De Carte's Rule of Signs. 2010-01-22. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/decartesruleofsigns.html.

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2010-01-22: Added "References" (McAdams, David.)
2009-01-22: Changed equations from images to html (McAdams, David.)
2008-12-04: Initial version (McAdams, David.)

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