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Watt's curve, which arose in the context of early work on the steam engine, is a sextic in two variables.. One method of solving the cubic equation involves transforming variables to obtain a sextic equation having terms only of degrees 6, 3, and 0, which can be solved as a quadratic equation in the cube of the variable.
Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (+) (), one can put it in standard form by expanding the products (by distributivity) and combining the like terms; for example, (+) = is of degree 1, even though each summand has ...
Since we study the computation of formal polynomials, we know that polynomials of very large degree require large circuits, for example, a polynomial of degree require a circuit of size roughly . So, the main goal is to prove lower bound for polynomials of small degree, say, polynomial in n . {\displaystyle n.}
Many circuit complexity classes are defined in terms of class hierarchies. For each non-negative integer i, there is a class NC i, consisting of polynomial-size circuits of depth ( ()), using bounded fan-in AND, OR, and NOT gates. The union NC of all of these classes is a subject to discussion.
A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results: = + +. This will exactly fit a simple curve to three points. If the order of the equation is increased to a ...
the multiplicative order, that is, the number of times the polynomial is divisible by some value; the order of the polynomial considered as a power series, that is, the degree of its non-zero term of lowest degree; or; the order of a spline, either the degree+1 of the polynomials defining the spline or the number of knot points used to ...
By the fundamental theorem of algebra, each polynomial of degree n must have n roots in the complex plane (i.e., for an ƒ with no roots on the imaginary line, p + q = n). Thus, we have the condition that ƒ is a (Hurwitz) stable polynomial if and only if p − q = n (the proof is given below).
Complex color plot of the Laguerre polynomial L n(x) with n as -1 divided by 9 and x as z to the power of 4 from -2-2i to 2+2i. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: ″ + ′ + =, = which is a second-order linear differential equation.