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Given a quadratic polynomial of the form + the numbers h and k may be interpreted as the Cartesian coordinates of the vertex (or stationary point) of the parabola. That is, h is the x -coordinate of the axis of symmetry (i.e. the axis of symmetry has equation x = h ), and k is the minimum value (or maximum value, if a < 0) of the quadratic ...
In a Church–Rosser system, an object has at most one normal form; that is the normal form of an object is unique if it exists, but it may well not exist. In lambda calculus for instance, the expression (λx.xx)(λx.xx) does not have a normal form because there exists an infinite sequence of β-reductions (λx.xx)(λx.xx) → (λx.xx)(λx.xx ...
For example, () is a rewrite rule, commonly used to establish a normal form with respect to the associativity of . That rule can be applied at the numerator in the term a ∗ ( ( a + 1 ) ∗ ( a + 2 ) ) 1 ∗ ( 2 ∗ 3 ) {\displaystyle {\frac {a*((a+1)*(a+2))}{1*(2*3)}}} with the matching substitution { x ↦ a , y ↦ a + 1 , z ↦ a + 2 ...
The quadratic trinomial in standard form (as from above): + + sum or difference of two cubes: = (+) A special type of trinomial can be factored in a manner similar to quadratics since it can be viewed as a quadratic in a new variable (x n below). This form is factored as:
This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials.
The general form of a linear equation with one variable, can be written as: + = Following the same procedure (i.e. subtract b from both sides, and then divide by a ), the general solution is given by x = c − b a {\displaystyle x={\frac {c-b}{a}}}
The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique up to the multiplication of the content by a unit of the ring of the coefficients (and the multiplication of the primitive part by the inverse of the ...
In algebra, a multilinear polynomial [1] is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables.