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Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers. For example, if a system contains 2 {\displaystyle {\sqrt {2}}} , a system over the rational numbers is obtained by adding the equation r 2 2 – 2 = 0 and replacing 2 {\displaystyle {\sqrt {2}}} by r 2 in the other equations.
Since ∧ and ¬ form a basis for Boolean algebra, all other logical operations are compositions of these basic operations, and so the polynomials of ordinary algebra can represent all Boolean operations, allowing Boolean reasoning to be performed using elementary algebra. For example, the Boolean 2-out-of-3 threshold or median operation is ...
This polynomial is further reduced to = + + which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing () and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were ...
If F is a field and f and g are polynomials in F[x] with g ≠ 0, then there exist unique polynomials q and r in F[x] with = + and such that the degree of r is smaller than the degree of g (using the convention that the polynomial 0 has a negative degree). The polynomials q and r are uniquely determined by f and g.
Solutions to polynomial systems computed using numerical algebraic geometric methods can be certified, meaning that the approximate solution is "correct".This can be achieved in several ways, either a priori using a certified tracker, [7] [8] or a posteriori by showing that the point is, say, in the basin of convergence for Newton's method.
Most general-purpose computer algebra systems have implementations of one or several algorithms for Gröbner bases, often also embedded in other functions, such as for solving systems of polynomial equations or for simplifying trigonometric functions; this is the case, for example, of CoCoA, GAP, Macaulay 2, Magma, Maple, Mathematica, SINGULAR ...
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction, , given above – manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability ...