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Trinomial Tree, Trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. Examples of trinomial expressions # 3x + 5y + 8z with x, y, z variables # 3t + 9s^2 + 3y^3 with t, s, y variables # 3ts + 9t + 5s with t, s variables # ax^2+bx+c, the quadratic polynomial in standard form with a,b,c variables.Quadratic expressions are not always trinomials, the expressions' appearance can vary. # A x^a y^b z^c + B t + C s with x, y, z, t, s variables, a, b, c nonnegative integers and A, B, C any constants. # Px^a + Qx^b + Rx^c where x is variable and constants a, b, c are nonnegative integers and P, Q, R any constants. Trinomial equation A trinomial equation is a polynomial equation involving three terms. An example is the equation x = q + x^m studied by Johann Heinrich Lambert in the 18th century. Some notable trinomials * The quadratic trinomial in standard form (as from above): :: ax^2+bx+c * sum or difference of two cubes: :: a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Pyramid Trinomial
Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French mathematician, physicist, inventor, philosopher, writer and theologian Places * Pascal (crater), a lunar crater * Pascal Island (Antarctica) * Pascal Island (Western Australia) Science and technology * Pascal (unit), the SI unit of pressure * Pascal (programming language), a programming language developed by Niklaus Wirth ** Microsoft Pascal **Turbo Pascal * PASCAL (database), a bibliographic database maintained by the Institute of Scientific and Technical Information * Pascal (microarchitecture), codename for a microarchitecture developed by Nvidia Other uses * (1895–1911) * (1931–1942) * Pascal and Maximus, fictional characters in ''Tangled'' * Pascal blanc, a French white wine grape * Pascal College, secondary education scho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruffini's Rule
In mathematics, Ruffini's rule is a method for computation of the Euclidean division of a polynomial by a binomial of the form ''x – r''. It was described by Paolo Ruffini in 1809. The rule is a special case of synthetic division in which the divisor is a linear factor. Algorithm The rule establishes a method for dividing the polynomial: :P(x)=a_nx^n+a_x^+\cdots+a_1x+a_0 by the binomial: :Q(x)=x-r to obtain the quotient polynomial: :R(x)=b_x^+b_x^+\cdots+b_1x+b_0. The algorithm is in fact the long division of ''P''(''x'') by ''Q''(''x''). To divide ''P''(''x'') by ''Q''(''x''): # Take the coefficients of ''P''(''x'') and write them down in order. Then, write ''r'' at the bottom-left edge just over the line: #: \begin & a_n & a_ & \dots & a_1 & a_0\\ r & & & & & \\ \hline & & & & & \\ \end # Pass the leftmost coefficient (''a''''n'') to the bottom just under the line. #: \begin & a_n & a_ & \dots & a_1 & a_0\\ r & & & & & \\ \hline & a_n & & & & \\ & =b_ & & & & \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Polynomial
In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, x^ + 3x^3 + 1 is a sparse polynomial, as it is a trinomial with a degree of 10. The motivation for studying sparse polynomials is to concentrate on the structure of a polynomial's monomials instead of its degree, as one can see, for instance, by comparing Bernstein–Kushnirenko theorem with Bezout's theorem. Research on sparse polynomials has also included work on algorithms whose running time grows as a function of the number of terms rather than on the degree, for problems including polynomial multiplication, division, root-finding algorithms, and polynomial greatest common divisors. Sparse polynomials have also been used in pure mathematics, especially in the study of Galois groups, because it has been easier to determine the Galois groups of certain families of sparse polynomials than it is f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Expression
In mathematics, an expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of operations. Expressions are commonly distinguished from ''formulas'': expressions are a kind of mathematical object, whereas formulas are statements ''about'' mathematical objects. This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, 8x-5 is an expression, while the inequality 8x-5 \geq 3 is a formula. To ''evaluate'' an expression means to find a numerical value equivalent to the expression. Expressions can be ''evaluated'' or ''simplified'' by replacing operations that appear in them with their result. For example, the expression 8\times 2-5 simplifies to 16-5, and evalua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Expression
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a primitive monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A primitive monomial is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multinomial (other)
Multinomial may refer to: * Multinomial theorem, and the multinomial coefficient * Multinomial distribution * Multinomial logistic regression * Multinomial test * Multi-index notation * Polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ... {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial (polynomial)
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials. Definition A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form :a x^m - bx^n , where and are numbers, and and are distinct non-negative integers and is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a ''Laurent binomial'', often simply called a ''binomial'', is similarly defined, but the exponents and may be negative. More generally, a binomial may be written as: :a\, x_1^\dotsb x_i^ - b\, x_1^\dotsb x_i^ Examples :3x - 2x^2 :xy + yx^2 :0.9 x^3 + \pi y^2 :2 x^3 + 7 Operations on simple binomials *The binomial , the difference of two squares, can be factored as the product of two other binomials: :: x^2 - y^2 = (x - y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a primitive monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A primitive monomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trinomial Expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by :(a+b+c)^n = \sum_ \, a^i \, b^ \;\! c^k, where is a nonnegative integer and the sum is taken over all combinations of nonnegative indices and such that . The trinomial coefficients are given by : = \frac \,. This formula is a special case of the multinomial formula for . The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron. Derivation The trinomial expansion can be calculated by applying the binomial expansion twice, setting d = b+c, which leads to : \begin (a+b+c)^n &= (a+d)^n = \sum_^ \, a^\, d^ \\ &= \sum_^ \, a^\, (b+c)^ \\ &= \sum_^ \, a^\, \sum_^ \, b^\,c^. \end Above, the resulting (b+c)^ in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index s. The product of the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization
In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''Factor (arithmetic), factors'', usually smaller or simpler objects of the same kind. For example, is an ''integer factorization'' of , and is a ''polynomial factorization'' of . Factorization is not usually considered meaningful within number systems possessing division ring, division, such as the real number, real or complex numbers, since any x can be trivially written as (xy)\times(1/y) whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by Greek mathematics, ancient Greek mathematicians in the case of integers. They proved the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Algebra
Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics), variables (quantities without fixed values). This use of variables entails use of algebraic notation and an understanding of the general rules of the Operation (mathematics), operations introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real number, real and complex numbers. It is typically taught to secondary school students and at introductory college level in the United States, and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization
In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''Factor (arithmetic), factors'', usually smaller or simpler objects of the same kind. For example, is an ''integer factorization'' of , and is a ''polynomial factorization'' of . Factorization is not usually considered meaningful within number systems possessing division ring, division, such as the real number, real or complex numbers, since any x can be trivially written as (xy)\times(1/y) whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by Greek mathematics, ancient Greek mathematicians in the case of integers. They proved the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
