Constant functionIn mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x (see image). As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c. Example: The function y(x) = 2 or just y = 2 is the specific constant function where the output value is c = 2. The domain of this function is the set of all real numbers R.
Rational root theoremIn algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer coefficients and . Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution x = p⁄q, written in lowest terms so that p and q are relatively prime, satisfies: p is an integer factor of the constant term a0, and q is an integer factor of the leading coefficient an.
Synthetic divisionIn algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as Ruffini's rule), but the method can be generalized to division by any polynomial. The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division.
Spline (mathematics)In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial (parametric) curve.
Substitution (logic)A substitution is a syntactic transformation on formal expressions. To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols with other expressions. The resulting expression is called a substitution instance, or instance for short, of the original expression. Where ψ and φ represent formulas of propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from φ by substituting formulas for symbols in φ, replacing each occurrence of the same symbol by an occurrence of the same formula.
Ruffini's ruleIn 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 1804. The rule is a special case of synthetic division in which the divisor is a linear factor. The rule establishes a method for dividing the polynomial: by the binomial: to obtain the quotient polynomial: 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.
Polynomial remainder theoremIn algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that, for every number any polynomial is the sum of and the product by of a polynomial in of degree less than the degree of In particular, is the remainder of the Euclidean division of by and is a divisor of if and only if a property known as the factor theorem. Let . Polynomial division of by gives the quotient and the remainder . Therefore, .
Algebraic fractionIn algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions. A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials. Thus is a rational fraction, but not because the numerator contains a square root function. In the algebraic fraction , the dividend a is called the numerator and the divisor b is called the denominator.
