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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 , − ∞ , {\displaystyle -\infty ,} and to introduce the arithmetic rules [ 9 ]
Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). [10] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero ...
A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined, [9] defined to be −1, [10] or defined to be a −∞. [11] A constant polynomial is either the zero polynomial, or a polynomial of degree zero.
Therefore, the polynomial does not have distinct roots if and only if it is divisible by the square of a polynomial of positive degree. This is the case if and only if the greatest common divisor of the polynomial and its derivative is not a constant. Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly ...
Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
A root of a polynomial is a zero of the corresponding polynomial function. [1] The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree , and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically ...
For example, a polynomial of degree n has a pole of degree n at infinity. The complex plane extended by a point at infinity is called the Riemann sphere . If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its ...