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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.
Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equation x 5 − 5x 4 + 30x 3 − 50x 2 + 55x − 21 = 0, for which the only real solution is
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial. [1] If F is any field, the trivial extension is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.
The polynomial 3x 2 − 5x + 4 is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two. [11]
If no such polynomial exists then the number is called transcendental. More generally the theory deals with algebraic independence of numbers. A set of numbers {α 1, α 2, …, α n} is called algebraically independent over a field K if there is no non-zero polynomial P in n variables with coefficients in K such that P(α 1, α 2, …, α n) = 0.
The equation of a line in a Euclidean plane is linear, that is, it equates a polynomial of degree one to zero. So, the Bézout bound for two lines is 1, meaning that two lines either intersect at a single point, or do not intersect. In the latter case, the lines are parallel and meet at a point at infinity.
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.