De Carte's rule of signs is a rule for determining the maximum number of positive roots of a polynomial.^{[1]} For any polynomial, the maximum number of positive roots 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 roots 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 nonzero term changes sign from the previous nonzero term, it counts as a sign change. Any roots that are neither negative or positive are complex roots.
Terms | Sign Changes | Positive Roots | f(-x) | Sign Changes | Negative Roots | Complex Roots |
---|---|---|---|---|---|---|
3x^{2} + 2 | 0 | 0 | 3x^{2} + 2 | 0 | 0 | 2 |
x - 2 | 1 | 1 | -x-2 | 0 | 0 | 0 |
3x^{2} + 2x + 7 | 0 | 0 | 3x^{2}-2x+7 | 2 | 2 or 0 | 0 or 2 |
-x^{3} - x^{2} + 5 | 1 | 1 | x^{3}-x^{2}+5 | 2 | 2 or 0 | 0 or 2 |
2x^{4} - 3x^{2} + 2 | 2 | 2 or 0 | 2x^{4}-3x^{2}+2 | 2 | 2 or 0 | 0, 2, or 4 |
4x^{6}-2x^{5}-3x^{4}-x^{3}+5x^{2}+4x-1 = 0 | 3 | 3 or 1 | 4x^{6}+2x^{5}-3x^{4}+x^{3}+5x^{2}-4x-1 = 0 | 3 | 3 or 1 | 2, 4, or 6 |
Table 1 |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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