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# algebraic equation

Mathematical statement of equality between algebraic expressions. An expression is algebraic if it involves a finite combination of numbers and variables and algebraic operations (addition, subtraction, multiplication, division, raising to a power, and extracting a root). Two important types of such equations are linear equations, in the form math.y = math.amath.x + math.b, and quadratic equations, in the form math.y = math.amath.x2 + math.bmath.x + math.c. A solution is a numerical value that makes the equation a true statement when substituted for a variable. In some cases it may be found using a formula; in others the equation may be rewritten in simpler form. Algebraic equations are particularly useful for modeling real-life phenomena.

In mathematics, an algebraic equation over a given field is an equation of the form
$P = Q$
where P and Q are (possibly multivariate) polynomials over that field. For example
$y^4+frac\left\{xy\right\}\left\{2\right\}=frac\left\{x^3\right\}\left\{3\right\}-xy^2+y^2-frac\left\{1\right\}\left\{7\right\}$
is an algebraic equation over the rationals.

Note that an algebraic equation over the rationals can always be converted to an equivalent one in which the coefficients are integers (where equivalence refers to the fact that the two equations will have the same solutions). For example, multiplying through by 42 = 2·3·7, the algebraic equation above becomes the algebraic equation

$42y^4+21xy=14x^3-42xy^2+42y^2-6$

Although the equation

$e^T x^2+frac\left\{1\right\}\left\{T\right\}xy+sin\left(T\right)z -2 =0$
is not an algebraic equation in four variables (x, y, z and T) over the rational numbers (because sine, exponentiation and 1/T are not polynomial functions) it is an algebraic equation in the three variables x, y, and z over Q((T)), the field of formal Laurent series in T over the rational numbers. Indeed, the coefficients

$e^T=1+T+frac\left\{T^2\right\}\left\{2!\right\}+frac\left\{T^3\right\}\left\{3!\right\}+cdots$
$sin\left(T\right)=T - frac\left\{T^3\right\}\left\{3!\right\} + frac\left\{T^5\right\}\left\{5!\right\} - frac\left\{T^7\right\}\left\{7!\right\} + cdots$

1/T and -2 are all elements of Q((T)).