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The degree of the zero polynomial is either left undefined, or is defined to be negative (usually −1 or ). [7] Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either.
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 polynomial, f(x) = 0, is the x-axis.
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.
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.
Then () is a polynomial of degree at most which has + distinct zeros (the ). But a non-zero polynomial of degree at most n {\displaystyle n} can have at most n {\displaystyle n} zeros, [ a ] so p ( x ) − q ( x ) {\displaystyle p(x)-q(x)} must be the zero polynomial, i.e. p ( x ) = q ( x ) {\displaystyle p(x)=q(x)} .
Here is a mathematics book that defines the degree of the zero polynomial to be -1: "By convention, the zero polynomial has degree -1." (p. 233) A Concrete Introduction to Higher Algebra By Lindsay N. Childs--50.53.50.168 04:39, 16 September 2014 (UTC)
Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial that ignores all its variables and always returns zero. The lemma states that evaluating a nonzero polynomial on inputs chosen randomly from a large-enough set is likely to find an input that produces a nonzero output.
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.