Difference between revisions of "Descartes Rule of Signs"
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| − | Descarte's rule of signs is a method used to determine the number of positive and negative roots of a polynomial. The rule gives an upper bound on the number of positive or negative roots, but does not specify the exact amount. | + | Descarte's rule of signs is a method used to determine the number of positive and negative roots of a polynomial. The rule gives an upper bound on the number of positive or negative roots, but does not specify the exact amount. |
| − | ==Positive roots== | + | ===Positive roots=== |
If the terms of a polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is equal to the number of sign differences between consecutive nonzero coefficients or is less than that by an even number. Multiple roots are counted separately. | If the terms of a polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is equal to the number of sign differences between consecutive nonzero coefficients or is less than that by an even number. Multiple roots are counted separately. | ||
| − | ==Negative roots== | + | ===Negative roots=== |
The bound for negative roots is a corollary of the positive root bound. The number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than that by a positive even number. | The bound for negative roots is a corollary of the positive root bound. The number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than that by a positive even number. | ||
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| + | ==See Also== | ||
| + | * [[Polynomials]] | ||
| + | * [[Factor Theorem]] | ||
| + | * [[Rational Root Theorem]] | ||
| + | *[[ Algebra/Intermediate]] | ||
| + | {{stub}} | ||
Latest revision as of 09:44, 20 September 2016
Descarte's rule of signs is a method used to determine the number of positive and negative roots of a polynomial. The rule gives an upper bound on the number of positive or negative roots, but does not specify the exact amount.
Positive roots
If the terms of a polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is equal to the number of sign differences between consecutive nonzero coefficients or is less than that by an even number. Multiple roots are counted separately.
Negative roots
The bound for negative roots is a corollary of the positive root bound. The number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than that by a positive even number.
See Also
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