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The propositions for the degree of sums and products of polynomials in the above section do not apply, if any of the polynomials involved is the zero polynomial. [8] It is convenient, however, to define the degree of the zero polynomial to be negative infinity, , and to introduce the arithmetic rules [9]
The largest zero of this polynomial which corresponds to the second largest zero of the original polynomial is found at 3 and is circled in red. The degree 5 polynomial is now divided by () to obtain = + + which is shown in yellow. The zero for this polynomial is found at 2 again using Newton's method and is circled in yellow.
Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none.
The number of positive real roots is at most the number of sign changes in the sequence of the polynomial's coefficients (omitting zero coefficients), and the difference between the root count and the sign change count is always even. In particular, when the number of sign changes is zero or one, then there are exactly zero or one positive roots.
An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n such that F(x) divides x n − 1 is n = p m − 1. A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1, meaning that any of them generates the multiplicative group ...
If one of the degrees is zero (that is, the corresponding polynomial is a nonzero constant polynomial), then there are zero rows consisting of coefficients of the other polynomial, and the Sylvester matrix is a diagonal matrix of dimension the degree of the non-constant polynomial, with the all diagonal coefficients equal to the constant ...
Also, even with a good approximation, when one evaluates a polynomial at an approximate root, one may get a result that is far to be close to zero. For example, if a polynomial of degree 20 (the degree of Wilkinson's polynomial) has a root close to 10, the derivative of the polynomial at the root may be of the order of ; this implies that an ...
where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on. Historically, polynomial models are among the most frequently used empirical models for curve fitting.