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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]
If a and b are rational numbers, the equation x 5 + ax + b = 0 is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ℓ and m such that
With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree greater than 100. [6] Computing the solutions in radicals of solvable polynomials requires huge computations. Even for the degree five, the expression of the solutions is so huge that it has no practical interest.
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.
If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q; If the degree of p is less than the degree of q, the limit is 0. If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur ...
Knot at infinity of X 0 (11) The classical modular curve, which we will call X 0 (n), is of degree greater than or equal to 2n when n > 1, with equality if and only if n is a prime. The polynomial Φ n has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with ...
In this case, the product of the degrees of the polynomials may be much larger than the number of roots, and better bounds are useful. Multi-homogeneous Bézout theorem provides such a better bound when the unknowns may be split into several subsets such that the degree of each polynomial in each subset is lower than the total degree of the ...
For triharmonic , () (where and are the weights and centers of ) is always a sum of total degree 5 polynomials in ,, and divided by the square root of a total degree 8 polynomial. Consider the behavior of these terms on the line x = a + t b {\displaystyle \mathbf {x} =\mathbf {a} +t\mathbf {b} } as t {\displaystyle t} approaches infinity.