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Trinomial

In mathematics, a trinomial is a polynomial consisting of exactly three terms, typically expressed in the form ax^2 + bx + c, where a, b, and c are constants and x is the variable. Trinomials may be of any degree, though quadratic forms (degree 2) are most common in basic algebra.^[1] These expressions are fundamental in algebra for operations such as addition, subtraction, multiplication, and factoring, often appearing in quadratic equations whose solutions reveal roots via methods like the quadratic formula or completing the square.^[2] Special cases include perfect square trinomials, which factor into the square of a binomial, such as (x + y)^2 = x^2 + 2xy + y^2.^[3] In biology, the term trinomial also refers to a three-part scientific name for a subspecies, such as Homo sapiens sapiens for modern humans.^[4]^[5]

Definition and Basic Concepts

Definition

A trinomial is a polynomial consisting of exactly three terms, where each term is a monomial formed by the product of a coefficient and one or more variables raised to non-negative integer powers.^[6] Unlike binomials or monomials, a trinomial requires precisely three distinct non-zero terms after combining any like terms, ensuring no reduction to fewer than three components.^[7] This distinguishes it within the broader category of polynomials, which may have any number of terms.^[8] The general form of a trinomial is ax^[m](/page/M+) + bx^[n](/page/N+) + cx^[p](/page/P′′), where a, b, and c are non-zero coefficients, and m, n, and p are non-negative integers with m > n > p to reflect descending order of exponents.^[8] This notation accommodates trinomials of varying degrees, though quadratic trinomials (where m = 2, n = 1, p = 0) are particularly common in elementary algebra.^[9] The term "trinomial" originates from the early modern period, combining the prefix "tri-" (from Latin trēs, meaning three) with "-nomial" (derived from Latin nōmen, meaning name, as in the terms of the expression), analogous to "binomial" for two terms.^[10] This etymology underscores the focus on the number of named components in the polynomial.^[11]

Terminology and Classification

In algebraic mathematics, a monomial is defined as a polynomial consisting of exactly one term, typically in the form ax^n where a is a coefficient and n is a non-negative integer exponent. A binomial extends this to precisely two such terms, while a trinomial is a polynomial with exactly three terms. The broader category of polynomials encompasses expressions with one or more terms, positioning the trinomial as a specific case distinguished by its term count.^[12]^[1] Trinomials are classified primarily by their degree, which represents the highest exponent (or total exponent sum in multivariable cases) among their terms. A linear trinomial has degree 1, consisting of three terms each of degree at most 1. A quadratic trinomial has degree 2 and takes the standard form ax^2 + bx + c with a \neq 0. Cubic trinomials possess degree 3, and those of higher degree (degree 4 or more) follow analogously, with the leading term determining the overall degree.^[1]^[13] Multivariable trinomials incorporate two or more variables, forming expressions with exactly three terms where each term may involve products of these variables raised to non-negative integer powers. For instance, forms like ax + by + cz illustrate linear multivariable trinomials, while higher-degree variants distribute the total degree across variables. This classification extends univariate concepts to higher dimensions, relevant in fields such as algebraic geometry.^[14]^[15] Trinomials are further categorized as homogeneous or non-homogeneous based on the uniformity of term degrees. A homogeneous trinomial requires all three terms to share the same total degree d, expressed as a linear combination of monomials each of degree d. In contrast, non-homogeneous trinomials feature terms with differing degrees, allowing mixed exponents as in the quadratic standard form. This distinction is fundamental in analyzing symmetries and scaling properties in polynomial systems.^[15]^[16]

Trinomial Expressions

Examples

Trinomials, as polynomials consisting of exactly three terms, can take various forms depending on their degree and the number of variables involved. These expressions are fundamental in algebra for demonstrating polynomial structure without like terms that would reduce the count below three.^[1] Simple examples include linear or low-degree trinomials such as x^2 + 2x + 1, which features three terms with powers of x and a constant; $3x + 5y + 7, a multivariable expression with distinct linear terms in x and y plus a constant; and a^3 + b^3 + c^3, involving cubic terms in three separate variables. Each qualifies as a trinomial because it comprises precisely three distinct monomials after any potential simplification, with no combining of like terms.^[17]^[18] Quadratic trinomials, typically of the form ax^2 + bx + c, illustrate the standard second-degree case. For instance, $4x^2 - 5x + 2 has a leading quadratic term, a linear term, and a constant, all distinct. The general form x^2 + rx + s similarly consists of three terms where r and s are coefficients, maintaining the trinomial structure as long as no terms combine. These examples highlight how quadratic trinomials appear in algebraic modeling and equation solving.^[19]^[20] Higher-degree trinomials extend beyond quadratics, such as x^4 + 3x^2 + 2, which includes even powers up to degree four, a quadratic term, and a constant—three distinct terms without simplification reducing the count. Another is $2x^3 + x^2 - 5x, featuring a cubic leading term, a quadratic term, and a linear term, all unlike. These demonstrate trinomials in advanced polynomial contexts, where the degree is determined by the highest power.^[21] Multivariable trinomials incorporate multiple variables, like x^2 + xy + y^2, with quadratic terms in x, a mixed xy term, and a y^2 term, forming three distinct monomials. Similarly, p^2 + 2pq + q^2 includes quadratic p, a product term pq, and quadratic q, qualifying as a trinomial due to its three non-combinable terms. Such forms are common in expressions involving two or more variables.^[18]

Operations and Simplification

Trinomials, as polynomials with exactly three terms, undergo the same algebraic operations as general polynomials, with simplification achieved by combining like terms to maintain or reduce to standard form. Addition of trinomials requires aligning like terms—those with identical variables and exponents—and summing their coefficients, resulting in another polynomial that may have up to three terms if no terms cancel. For instance, adding x^2 + 2x + 1 and $3x^2 - x + 4 yields (x^2 + 3x^2) + (2x - x) + (1 + 4) = 4x^2 + x + 5, preserving the trinomial structure.^[1] Subtraction follows a similar process after distributing a negative sign to the subtrahend, such as subtracting $2x^2 + x - 3 from x^2 + 3x + 2 to obtain (x^2 - 2x^2) + (3x - x) + (2 - (-3)) = -x^2 + 2x + 5.^[1] Multiplication of a trinomial by a monomial involves distributing the monomial across each term of the trinomial and simplifying the resulting expression, which may produce a polynomial of higher degree. For example, multiplying x^2 + 2x + 1 by $3x gives $3x \cdot x^2 + 3x \cdot 2x + 3x \cdot 1 = 3x^3 + 6x^2 + 3x. When multiplying a trinomial by a binomial, the distributive property (often using the FOIL method extended to three terms) is applied repeatedly, potentially yielding a quartic polynomial; consider (x^2 + 2x + 1)(x + 1) = x^3 + x^2 + 2x^2 + 2x + x + 1 = x^3 + 3x^2 + 3x + 1.^[1] Simplification of trinomials after operations entails identifying like terms, combining their coefficients, and arranging the result in descending order of exponents to achieve standard form, ensuring no more than three terms unless the operation increases the degree. This process eliminates redundant expressions and verifies the polynomial's structure, such as reducing $4x^2 + x + 5 - (x^2 - 2x + 1) = 3x^2 + 3x + 4 . If the result has fewer than three terms, it is no longer a trinomial but a simplified binomial or monomial.^[22] Division of a trinomial by a monomial divides each term's coefficient and variable separately, producing a rational expression that simplifies to a polynomial only if the monomial divides evenly into each term without remainder. For basic cases, rewrite the trinomial as a sum of fractions with the monomial in the denominator, then simplify; dividing $6x^3 + 3x^2 - 9x by $3x yields \frac{6x^3}{3x} + \frac{3x^2}{3x} - \frac{9x}{3x} = 2x^2 + x - 3. Remainders occur if division is inexact, such as x^2 + 2x + 1 divided by $2x resulting in \frac{x^2}{2x} + \frac{2x}{2x} + \frac{1}{2x} = \frac{x}{2} + 1 + \frac{1}{2x}, leaving a rational term.^[23]

Factoring Trinomials

Perfect Square Trinomials

A perfect square trinomial is a quadratic polynomial that factors perfectly into the square of a binomial, taking the form a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2, where a and b are algebraic expressions.^[24] This structure arises directly from the binomial square identity and represents a special case of quadratic trinomials where the expression is a complete square.^[25] To recognize a perfect square trinomial, examine the coefficients: the absolute value of the middle term must equal twice the product of the square roots of the leading and constant terms.^[26] For instance, in x^2 + 6x + 9, the square root of the first term is x and of the last is 3, so twice their product is $2 \cdot x \cdot 3 = 6x, matching the middle term.^[27] This criterion allows quick identification without full expansion or factoring. The expansion of a perfect square trinomial derives from the binomial theorem applied to (a \pm b)^2, yielding a^2 \pm 2ab + b^2.^[25] For example, expanding (x + 3)^2 gives x^2 + 2 \cdot x \cdot 3 + 3^2 = x^2 + 6x + 9, illustrating how the middle term is precisely twice the product of the outer terms.^[24] Similarly, (x - 4)^2 = x^2 - 8x + 16 follows the same pattern with a negative middle coefficient.^[27] Verification of a perfect square trinomial can be confirmed using the discriminant of the quadratic ax^2 + bx + c, where a discriminant of zero (b^2 - 4ac = 0) indicates the quadratic has a repeated root and factors as a perfect square.^[28] For x^2 + 6x + 9, the discriminant is $6^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0, confirming it as (x + 3)^2.^[29] This method provides an algebraic check, especially useful when coefficients are not immediately obvious.^[30]

General Quadratic Trinomials

A general quadratic trinomial takes the form ax^2 + bx + c, where a, b, and c are constants with a \neq 0, and the expression does not factor as a perfect square.^[31] Unlike perfect square trinomials, which have a discriminant of zero, general quadratic trinomials typically require systematic methods to decompose into linear factors when possible, assuming integer or rational coefficients.^[31] These methods rely on the trinomial's factorability over the integers or rationals, determined by whether suitable factor pairs exist.^[32] When a = 1, a trial-and-error approach, often called the product-sum method, identifies two integers that multiply to c and add to b, allowing the trinomial to be expressed as (x + m)(x + n).^[32] For example, consider x^2 + 5x + 6: the factors 2 and 3 of 6 add to 5, yielding (x + 2)(x + 3).^[32] If a \neq 1, the AC method extends this by first computing the product ac and finding factor pairs that sum to b; these are then used to rewrite the middle term, enabling grouping and factoring into binomials.^[32] For instance, $2x^2 + 7x + 3 has ac = 6, with factors 1 and 6 summing to 7; rewriting gives $2x^2 + x + 6x + 3 = (2x + 1)(x + 3).^[32] Alternative techniques include completing the square, which rewrites the trinomial by adding and subtracting \left( \frac{b}{2a} \right)^2 (after adjusting for a = 1) to form a perfect square plus a constant, aiding in algebraic manipulation or solving.^[33] The quadratic formula, x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, provides the roots directly, allowing factorization as a(x - r_1)(x - r_2) when the roots are real and rational.^[31] However, if the discriminant b^2 - 4ac < 0, no real roots exist, rendering the trinomial irreducible over the real numbers.^[31]

Trinomial Equations

Quadratic Trinomial Equations

A quadratic trinomial equation is a polynomial equation of degree two in which the polynomial is a trinomial, typically expressed in the standard form ax^2 + bx + c = 0, where a, b, and c are constants and a \neq 0.^[34]^[35] This form represents equations that arise in various mathematical contexts, such as modeling projectile motion or optimizing areas, and solving them involves finding the values of x that satisfy the equation.^[36] Several methods exist for solving quadratic trinomial equations. One approach is factoring, where the trinomial is decomposed into a product of linear factors, leveraging the roots identified through prior factorization techniques to apply the zero-product property.^[37]^[38] Another method is completing the square, which involves rewriting the equation by adding and subtracting a constant to form a perfect square trinomial, allowing isolation of the variable through square roots.^[36]^[39] The most general and versatile method is the quadratic formula, derived from completing the square, given by

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},

which provides the roots explicitly for any coefficients a, b, and c.^[30]^[40] The discriminant, defined as d = b^2 - 4ac, plays a crucial role in determining the nature of the roots. If d > 0, there are two distinct real roots; if d = 0, there is exactly one real root (a repeated root); and if d < 0, there are no real roots, only complex conjugate roots.^[41]^[42]^[43] This quantity influences the selection of solution method and the equation's solvability over the real numbers. For example, consider the equation x^2 - 5x + 6 = 0. Factoring yields (x - 2)(x - 3) = 0, so the solutions are x = 2 and x = 3.^[38]^[44] Here, the discriminant is d = (-5)^2 - 4(1)(6) = 1 > 0, confirming two real roots. Graphically, the solutions to a quadratic trinomial equation correspond to the x-intercepts of the parabola y = ax^2 + bx + c, where the number and position of these intersections with the x-axis reflect the roots determined algebraically.^[45]^[46]^[47]

Higher-Degree Trinomial Equations

Higher-degree trinomial equations involve polynomials of degree three or more with exactly three terms set equal to zero. Common forms include the depressed cubic equation x^3 + px + q = 0, where the quadratic term is absent, and biquadratic trinomials of the form x^4 + rx^2 + s = 0. These equations often require specialized techniques beyond simple factoring, as they do not generally factor into linear terms over the reals.^[48]^[49] For even-degree trinomial equations like biquadratics, a substitution method reduces the problem to a quadratic equation. Let y = x^2, transforming x^4 + rx^2 + s = 0 into y^2 + ry + s = 0. Solve this quadratic for y using the quadratic formula, then back-substitute by solving x^2 = y for each real positive root of y, yielding x = \pm \sqrt{y}. If y is negative, the roots are complex: x = \pm i \sqrt{|y|}. For example, consider x^4 + 5x^2 + 4 = 0. Substituting y = x^2 gives y^2 + 5y + 4 = 0, which factors as (y + 1)(y + 4) = 0, so y = -1 or y = -4. Thus, x^2 = -1 implies x = \pm i, and x^2 = -4 implies x = \pm 2i. All roots are complex in this case.^[49] Cubic trinomials in depressed form are solved using Cardano's formula, a radical-based method developed in the 16th century. For x^3 + px + q = 0, the real root is given by

x = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{\frac{-q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{\frac{-q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}},

provided the discriminant \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 \geq 0; otherwise, trigonometric or hyperbolic forms are used for three real roots. This formula, while exact, is complex for computation, leading to numerical methods like Newton-Raphson iteration for approximations when analytical solutions are impractical. For irreducible cases over the rationals, these approximations establish root locations without explicit radicals.^[48] Historically, Johann Heinrich Lambert addressed trinomial equations of the form x + x^m = q in 1758 by developing a power series expansion for x in terms of q, providing an early analytical approach to such higher-degree problems. This work laid groundwork for later functions like the Lambert W function, though it focused on series convergence rather than closed forms.^[50]

Special and Notable Trinomials

Mathematical Identities

One prominent mathematical identity involving trinomials arises in the factorization of sums and differences of cubes. The sum of cubes formula states that a^3 + b^3 = (a + b)(a^2 - ab + b^2), where a^2 - ab + b^2 is a trinomial factor that cannot be further factored over the real numbers.^[51] Similarly, the difference of cubes is given by a^3 - b^3 = (a - b)(a^2 + ab + b^2), with a^2 + ab + b^2 serving as the trinomial factor.^[52] These identities are derived from the binomial expansion of (a + b)^3 and (a - b)^3, respectively, and are fundamental for simplifying higher-degree expressions in algebra.^[53] Another key identity for trinomials appears in the factorization of polynomials that are quadratic in terms of x^n. Specifically, an expression of the form x^{2n} + r x^n + s factors as (x^n + p)(x^n + q), where p and q satisfy p + q = r and p q = s.^[54] This approach treats x^n as a single variable u, reducing the problem to factoring the quadratic u^2 + r u + s, and is particularly useful for even-degree polynomials where direct factoring is challenging. For instance, when n = 2, x^4 + 5 x^2 + 6 = (x^2 + 2)(x^2 + 3), illustrating the trinomial intermediate step in the quadratic form.^[55] Extensions of the binomial theorem to trinomials are captured by the multinomial theorem, which provides a general expansion for (a + b + c)^n and allows isolation of trinomial patterns in partial terms.^[56] For n = 2, the expansion (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc yields a symmetric expression emphasizing cross-term contributions like $2ab + 2ac + 2bc.^[57] This identity facilitates manipulations in multivariable algebra and symmetric polynomials.

Applications and Examples in Mathematics

In calculus, the second-order Taylor polynomial provides a quadratic trinomial approximation to a smooth function f near a point a, given by

P_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2.

This trinomial captures the function's value, slope, and concavity at a, enabling local approximations essential for error analysis, numerical integration, and optimization algorithms.^[58] In probability theory, the trinomial distribution extends the binomial model to independent trials with three outcomes, probabilities p, q, r (summing to 1), and n trials. The joint probability mass function for counts X = k, Y = m, Z = n - k - m is

P(X = k, Y = m) = \frac{n!}{k! \, m! \, (n - k - m)!} p^k q^m r^{n - k - m},

where the trinomial coefficient \frac{n!}{k! \, m! \, (n - k - m)!} generalizes binomial coefficients and appears in applications like three-state Markov chains or trinomial trees in finance.^[59] In number theory, trinomials feature prominently in Diophantine equations via binary quadratic forms ax^2 + bxy + cy^2 = N (trinomial when b \neq 0), which generalize Pell's equation x^2 - dy^2 = 1 (the case b = 0). Solutions to these equations, determining integer representations of N, rely on automorphism groups of the form and solutions to associated Pell-like equations, as in the study of class numbers and units in quadratic fields. For example, the form x^2 + xy + y^2 corresponds to the norm equation in the Eisenstein integers \mathbb{Z}[\omega], where \omega is a primitive cube root of unity.^[60] Leonhard Euler advanced the theory of trinomial expansions in his 1765 paper Observationes analyticae (Eneström number E326), expressing coefficients of (1 + x + x^2)^n as alternating sums of binomial coefficients and exploring general multinomial cases. This contributed to early developments in generating functions and hypergeometric series. A seminal example is the trinomial x^2 + x + 1 = \Phi_3(x), the third cyclotomic polynomial, which is irreducible over \mathbb{Q} and generates the cyclotomic field \mathbb{Q}(\zeta_3) for the primitive cube root of unity \zeta_3.^[61]^[62] In geometry, trinomials model quadratic expressions for areas and volumes; for instance, in derivations of Heron's formula A = \sqrt{s(s - a)(s - b)(s - c)} using the law of cosines, the area A = \frac{1}{2}ab \sin C substitutes \cos C = \frac{a^2 + b^2 - c^2}{2ab}, yielding \sin^2 C = 1 - \cos^2 C as a quadratic trinomial in the squared sides that simplifies to the Heron radicand after algebraic manipulation.^[63]

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[PDF] The Quadratic Formula and the Discriminant
If the discriminant is greater than zero, there will be two real solutions as in examples one and two.
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Algebra - Quadratic Equations - Part I - Pauls Online Math Notes
Nov 16, 2022 · Quadratic equations are in the form ax^2 + bx + c = 0. This section covers solving by factoring and the square root property.
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Parabolas - CSUN
The graph of a quadratic equation in two variables (y = ax2 + bx + c ) is called a parabola. The following graphs are two typical parabolas their x-intercepts ...
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2-02 Quadratic Equations
The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros , or roots , of the quadratic ...
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[PDF] Parabolas
If the quadratic equation ax2 + bx + c = 0 has two roots, then the parabola intersects the x-axis at two points. If the equation has one root, then the parabola ...
[49]
Cardano's Method | Brilliant Math & Science Wiki
The solution has two steps. We first "depress" the cubic equation and then solve the depressed equation.
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10.5 Solving Quadratic Equations Using Substitution
Substitution involves converting the equation to a more common form by substituting, then factoring, and replacing the substitutions with original variables.
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[PDF] On the Lambert W Function - London - Western University
In 1758, Lambert solved the trinomial equation x = q +xm by giving a series develop- ment for x in powers of q. Later, he extended the series to give powers of ...
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[PDF] college algebra sum of cubes | Bluefield Esports
Q: What is the sum of cubes formula in college algebra? A: The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. This identity allows you to factor ...
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[PDF] Sum & Difference of Cubes
A polynomial in the form a 3 + b 3 is called a sum of cubes. A polynomial in the form a 3 – b 3 is called a difference of cubes.
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[PDF] Algebraic Manipulations | MIT
Expand the following expressions: (a) (a + b)2. (b) (a - b)2. (c) (a - b)(a + b). (d) (a + b)3. (e) (a - b)3. (f) (a + b)(a2 - ab + b2). (g) (a - b)(a2 + ab + ...
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[PDF] Factoring Polynomials in Quadratic Form | LAVC
Factoring Polynomials in Quadratic Form. Factor each completely. 1) x. 4 + 2x. 2 − 63. 2) a. 4 + 14a. 2 + 40. 3) 4x. 4 − 48x. 2 + 128. 4) 2m. 4 − 36m.
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Algebra - Factoring Polynomials - Pauls Online Math Notes
Nov 16, 2022 · Factoring polynomials involves determining what multiplied to get the polynomial, then factoring each term until completely factored. Methods ...
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[PDF] Discrete Math for Computer Science Students
Trinomial coefficient. We define the trinomial coefficient ( n k1,k2,k3) to be n! k1!k2!k3! if k1 +k2 + k3 = n and 0 otherwise. 6. Trinomial Theorem. The ...
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[PDF] REVIEW OF ALGEBRA - Stewart Calculus
We have used the Distributive Law to expand certain algebraic expressions. We sometimes need to reverse this process (again using the Distributive Law) by ...<|control11|><|separator|>
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Calculus II - Taylor Series - Pauls Online Math Notes
Nov 16, 2022 · The Taylor Series for f(x) about x=a is f(x)=∞∑n=0f(n)(a)n!(x−a)n, where cn=f(n)(a)/n!. If a=0, it's called a Maclaurin Series.
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17.3 - The Trinomial Distribution | STAT 414 - STAT ONLINE
What happens if there aren't two, but rather three, possible outcomes? That's what we'll explore here on this page, ending up not with the binomial distribution ...Missing: multinomial expansion<|control11|><|separator|>
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[PDF] Binary Quadratic Forms - UCSD Math
If f(x, y) = x2 −dy2, then asking if the integer 1 is represented by f (nontrivially) is the same as asking if Pell's equation has a nontrivial solution.
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Trinomial Coefficient -- from Wolfram MathWorld
Trinomial coefficients have a rather sparse literature, although no less than Euler (in 1765) authored a 20-page paper on their properties (Andrews 1990). The ...
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Cyclotomic Polynomial -- from Wolfram MathWorld
Cyclotomic Polynomial ; = x^(p-1)+x^(p-2)+ ; Phi_(2p)(x), = (x^(2p)-1)/(x^p-1 ; = x^(p-1)-x^(p-2)+ ; Phi_(4p)(x), = (x^(4p)-1)/(x^(2p) ; = x^(2p-2)-x^(2p-4)+.
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[PDF] A simple proof of Heron's formula for the area of a triangle
Jul 22, 2020 · The area of a triangle with side lengths a, b, c is equal to (1) A(a, b, c) = p s(s - a)(s - b)(s - c),