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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.
Every polynomial in one variable x with real coefficients can be uniquely written as the product of a constant, polynomials of the form x + a with a real, and polynomials of the form x 2 + ax + b with a and b real and a 2 − 4b < 0 (which is the same thing as saying that the polynomial x 2 + ax + b has no real roots).
This works well for every degree, but, in degrees higher than four, the resulting polynomial that has the s i as roots has a degree higher than that of the initial polynomial, and is therefore unhelpful for solving. This is the reason for which Lagrange's method fails in degrees five and higher.
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.
Its existence is based on the following theorem: Given two univariate polynomials a and b ≠ 0 defined over a field, there exist two polynomials q (the quotient) and r (the remainder) which satisfy = + and < (), where "deg(...)" denotes the degree and the degree of the zero polynomial is defined as being negative.