cubic

From Middle French cubique, equivalent to cube + -ic.


quadratic

Trésor de la langue française informatisé

From French quadratique (1765), from Latin quadrātus + -ique (English -ic), form of quadrō (“I make square”), from quādrus (“square”), ultimately from Proto-Indo-European *kʷetwóres (“four”), whence also Latin quartus (“fourth”).

(mathematics) Of a polynomial, equation or function, second-degree: involving the second power (square) of a variable but no higher powers, as ax²+bx+c.

քառակուսի

of a class of polynomial of the form y = ax² + bx + c


linear

en-adj

linear (comparative more linear, superlative most linear)

more linear (comparative)
most linear (superlative)

From Latin līneāris, from līnea (“line”) + -āris (adjectival suffix), equivalent to line + -ar. Doublet of lineal.

lineêr

in mathematics, of first-degree polynomial

4x#x2B;5y#x2B;#x5C;frac#x7B;1#x7D;#x7B;2#x7D;z is a linear polynomial, but 4xy#x2B;#x5C;frac#x7B;1#x7D;#x7B;2#x7D;z and 4x#x2B;z² are not.

(mathematics) (of polynomials or polynomial equations)

(of a polynomial) Having degree less than one; that is, being of the form a_1x_1+a_2x_2+⋯+a_nx_n, where each x_i is a variable and each a_i is a coefficient. See also Linear polynomials on Wikipedia.Wikipedia

The graph of the linear equation y#x3D;ax#x2B;b is a straight line with slope a and y-intercept b

(of a polynomial equation) Involving only linear polynomials. See also Linear equation on Wikipedia.Wikipedia


linear

linear amplifier
linear amplifier#English

(radio slang) Ellipsis of linear amplifier..


binomial

Formed from Late Latin binōmium + -al. The derivation of binōmium is unclear. It was used by Gerard of Cremona in the 12th century. Suggested sources are the Latin nōmen (“name”), the Ancient Greek νομός (nomós, “distribution, pasture”), or the Old French nom (“name”). Compare binomy and binominal, as well as the French binôme. By surface analysis, bi- + -nomial.


polynomial

/ˌpɒlɪˈnəʊmɪəl/

From poly- + -nomial, from νομός (nomós, “portion, part”), by analogy with binomial.

(algebra, strict sense) An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a_nxⁿ+a_n-1xⁿ⁻¹+...+a_0x⁰.


cubic curve

(mathematics) A plane curve having the equation y=a.x³+b.x²+c.x+d.


octic

Unfortunately, an analogous attack on the octic polynomial becomes exceedingly complicated.

1999, William Dunham, Euler: The Master of Us All, Mathematical Association of America, published 1999, page 116:

(mathematics) Of the eighth degree or order.


septic

Enriques states that it is possible to construct a family of septics with a singular curve of degree 7 and genus 4 having a triple point that degenerates to the above configuration[…].

2002, Ingrid C. Bauer, Fabrizio Catanese, Roberto Pignatelli, “Canonical Rings of Surfaces Whose Canonical System has Base Points”, in Ingrid C. Bauer et al., editors, Complex Geometry: Collection of Papers Dedicated to Hans Grauert, page 66:

Now consider the two septics C = U⁷ᵢ₌₁Cᵢ, D = U⁷ᵢ₌₁Dᵢ and note that for i = 1,2,3,4,5,6 the lines Cᵢ and Dᵢ are parallel, so that the intersection of two septics S' consists of 6⁶^([sic]) + 6 + 1 = 43 points and it is the singular set of a foliation of degree 6.

2003, Antonio Campillo, Santiago Encinas, “Two Dimensional Complete Ideals”, in Luchezar L. Avramov et al., editors, Commutative Algebra: Interactions with Algebraic Geometry: International Conference, page 71:

(mathematics) A mathematical object (function, curve, surface, etc.) of degree seven.


nonic

(mathematics) Describing a polynomial that has 9 as the highest exponent of its terms.


polynomial equation

en-noun

polynomial equation (plural polynomial equations)

(algebra) Any algebraic equation in which one or both sides are in the form of a polynomial.


cyclotomic

(mathematics) of, or relating to the complex roots of unity


quartic

LL-Q1860 (eng)-Flame, not lame-quartic.wav

(mathematics) An algebraic equation or function of the fourth degree.


quantic

When the two quantics are the first derived functions of the same quantic of any odd order, the lineo-linear invariant does not vanish, but it is not an invariant of the single quantic.

1858, Arthur Cayley, “A Fourth Memoir on Quantics”, in Philosophical Transactions of the Royal Society of London, volume 148, published 1859, page 421:

74. The discriminant of a binary quantic, or the eliminant of a system of binary quantics, is an invariant.
We can see a priori that this must be the case, for if a given quantic has a square factor, it will have a square factor still when it is linearly transformed; or if a system of quantics have a common factor, they will still have a common factor when the equations are transformed.

1859, George Salmon, Modern Higher Algebra, page 52:

homogeneous polynomial
homogeneous polynomial

(mathematics) A homogeneous polynomial in two or more variables.


quintic

LL-Q1860 (eng)-Flame, not lame-quintic.wav

From Latin quintus (“fifth”) + -ic (“of pertaining to”).

(mathematics) Of or relating to the fifth degree, such as a quintic polynomial which has the form ax⁵+bx⁴+cx³+dx²+ex+f=0 (containing a term with the independent variable raised to the fifth power).


quadratic equation

quadratic equation (plural quadratic equations)

When I was seven years old at a new school, I cried all day, every day. When I was 14 years old, I cried in maths lessons. An improvement, I guess. It wasn’t that I was bad at maths; I knew my way around a quadratic equation, let me tell you.

2015 September 8, Katie Everson, “Why I've written a book for teenagers about taking drugs”, in The Guardian:

The arid argot of the estate agent’s schedule becomes obsolete when the task is to sell a pub-restaurant at the edge of a Highland wilderness. It’s like trying to describe the glory of the Lake District with a quadratic equation.

2018 November 25, Kevin McKenna, “Remote island pub for sale ... just don’t expect a quiet life”, in The Guardian:

The quadratic equation has frustrated math students for millenniums. […] If, while watching the Super Bowl, you had wanted to estimate how far a pass thrown by Patrick Mahomes traveled through the air, you would have been solving a quadratic equation.

2020 February 5, Kenneth Chang, Jonathan Corum, “This Professor’s ‘Amazing’ Trick Makes Quadratic Equations Easier”, in The New York Times:

If there’s one skill we absolutely should have been taught in school, it's how to craft the perfect dating app profile. I mean, if I can still remember the quadratic equation, surely I should have been schooled in the art of snagging a date. But alas, our education system seems to think finding our soulmates isn't as essential as algebra.

2024 April 8, Lydia Wang, Morgan Sullivan, Rachel Varina, “24 Best Hinge Prompts to Actually Help Your Matches Get to Know You”, in Cosmopolitan:

(algebra) A polynomial equation of the second degree; an equation that can be rearranged in standard form as ax²+bx+c=0, where x is an unknown value, and a, b, and c are known numbers, where a does not equal zero.


monic

(mathematics) Of a polynomial whose leading coefficient is one.


quadratic formula

quadratic formula (plural quadratic formulas or quadratic formulae)

quadratic formulas (plural)
quadratic formulae (plural)

(mathematics) The formula x=(-b±√)/(2a) for finding roots of the quadratic equation.

toisen asteen yhtälön ratkaisukaava

the formula for finding roots of the quadratic equation


linear function

linear function (plural linear functions)

(mathematics) Any function whose graph is a straight line: f(x)=ax+b


power function

independent variable
independent variable

(mathematics) A function where the independent variable is raised to a particular real number power.


quadratic function

quadratic function (plural quadratic functions)

(mathematics) Any function whose value is the solution of a quadratic polynomial


cubic function

(mathematics) Any function (polynomial) whose greatest exponent is 3.


cubic equation

(mathematics) A polynomial equation whose greatest exponent is 3.


homogeneous polynomial

homogeneous polynomial (plural homogeneous polynomials)

(mathematics) a polynomial such that the sum of the exponents of the variables is the same for every term.

齐次多项式

mathematics: polynomial such that the sum of the exponents of the variables is the same for every term


symmetric polynomial

symmetric polynomial (plural symmetric polynomials)

x+y+z

xyz

2x²+2y²+2z², etc.

An example of a non-symmetric polynomial is x+2y: if we interchange x and y, this polynomial changes to 2x+y.

(algebra) A polynomial which remains unchanged if its variables are permuted


elementary symmetric polynomial

elementary symmetric polynomial (plural elementary symmetric polynomials)

elementary symmetric polynomials (plural)

(algebra) a coefficient of some power of t in (t+x_1)(t+x_2)⋯(t+x_n), for some n∈ N


monomial

From mononomial, with -nono- simplified to -no- by haplology.

5x²y is a monomial

(mathematics) A single term consisting of a product of numbers and variables with positive integer exponents.


trinomial

(algebra) An expression consisting of three terms.


quadrinomial

(algebra) An expression consisting of four terms.


quintic function

quintic function (plural quintic functions)

(mathematics) Any function of a polynomial whose greatest exponent is 5.


quartic function

quartic function (plural quartic functions)

(mathematics) Any function of a polynomial whose greatest exponent is 4.


sextic

(mathematics) An algebraic equation or function of the sixth degree, e.g. a polynomial of degree 6; a curve surface etc. describing such an equation or function, especially a quantic of the sixth degree.


decic

(mathematics) Of the tenth degree or order; e.g. for a polynomial


quintinomial

(mathematics) Any expression that has five terms.


quadratic form

Quadratic form
Quadratic form (statistics)

Historically, quadratic form theory has been treated as a rich but misunderstood uncle.

1980, John Gregory, Quadratic Form Theory and Differential Equations, Academic Press, page ix:

2004, Nikita A. Karpenko, Izhboldin's Results on Stably Birational of Equivalence Quadrics, Oleg T. Izhboldin, Geometric Methods in the Algebraic Theory of Quadratic Forms, Springer, Lecture Notes in Mathematics 1835, page 156,
We claim that this quadratic form is isotropic, and this gives what we need according to [2, Proposition 4.4].

2009, R. Parimala, V. Suresh, J.-P. Tignol, On the Pfister Number of Quadratic Forms, Ricardo Baeza, Quadratic Forms—Algebra, Arithmetic, and Geometry, American Mathematical Society, page 327,
The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of −1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n − 2 terms and not less. The number of Pfister forms required to express a quadratic form of degree 6 with trivial discriminant is determined in various cases.

The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange's theorem that every positive integer is a sum of four squares, was continued in the 19th century by Dirichlet, Hermite, H. J. S. Smith, Minkowski and others.

2009, W. A. Coppel, Number Theory: An Introduction to Mathematics, 2nd edition, Springer, page 291:

(mathematics, number theory, algebra) A homogeneous polynomial of degree 2 in a given number of variables.


polynomial form

polynomial form (plural polynomial forms)

(algebra) A linear combination of powers of an indeterminate (or products of powers of more than one indeterminate), with coefficients belonging to an integral domain or a field. (The indeterminate is thought of as an element extraneous to the set of coefficients, instead of as a variable element of it (as in the case of polynomial functions), just as, say, the square root of negative one is an element extraneous to the set of integers when it is adjoined to them to form the domain of Gaussian integers. The indeterminate forms a free commutative monoid, to which all powers of it belong, and the unity of it can also show up implicitly in the constant term of a polynomial form.)


cyclotomic polynomial

cyclotomic polynomial (plural cyclotomic polynomials)

For a prime number p, the pᵗʰ cyclotomic polynomial is xᵖ-1/x-1=xᵖ⁻¹+xᵖ⁻²+...+x²+x+1.

Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing x with x+1.

Euler's totient function
Euler's totient function

(algebra) For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial.


Legendre polynomial

Legendre polynomial (plural Legendre polynomials)

Named after Adrien-Marie Legendre, who discovered them in 1782.

(mathematics) Any of a certain system of complete and orthogonal polynomials, with numerous mathematical properties and applications.


Jacobi polynomial

Jacobi polynomial (plural Jacobi polynomials)

(mathematics) Any member of a certain class of orthogonal polynomials.

Carl Gustav Jacob Jacobi
Jacobi polynomial