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In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
Xcas has the ability of a scientific calculator that provides show input and writes pretty print; Xcas works also as a spreadsheet; [11] computer algebra; 2D geometry in the plane; [12] 3D geometry in space; [12] spreadsheet; [11] statistics; regression (exponential, linear, logarithmic, logistic, polynomial, power) programming; [13]
In algebra, a quartic function is a function of the form = + + + +, α. where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r 1 and r 2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
Putting a formula into ANF also makes it easy to identify linear functions (used, for example, in linear-feedback shift registers): a linear function is one that is a sum of single literals. Properties of nonlinear-feedback shift registers can also be deduced from certain properties of the feedback function in ANF.
the linear mapping : makes a cyclic []-module, having a basis of the form {,, …,}; or equivalently [] / (()) as []-modules. If the above hold, one says that A is non-derogatory . Not every square matrix is similar to a companion matrix, but every square matrix is similar to a block diagonal matrix made of companion matrices.
On the other hand, this makes the Frobenius normal form rather different from other normal forms that do depend on factoring the characteristic polynomial, notably the diagonal form (if A is diagonalizable) or more generally the Jordan normal form (if the characteristic polynomial splits into linear factors). For instance, the Frobenius normal ...
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