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{{main|Glossary of invariant theory}}
A binary form (of degree ''n'') is a homogeneous polynomial Σ{{su|b=''i''=0|p=''n''}} ({{su|p=''n''|b=''i''}})''a''<sub>''n''−''i''</sub>''x''<sup>''n''−''i''</sup>''y''<sup>''i''</sup> = ''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ({{su|p=''n''|b=1}})''a''<sub>''n''−1</sub>''x''<sup>''n''−1</sup>''y'' + ... + ''a''<sub>0</sub>''y''<sup>''n''</sup>. The group ''SL''<sub>2</sub>('''C''') acts on these forms by taking ''x'' to ''ax'' + ''by'' and ''y'' to ''cx'' + ''dy''. This induces an action on the space spanned by ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> and on the polynomials in these variables. An '''invariant''' is a polynomial in these ''n'' + 1 variables ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> that is invariant under this action. More generally a '''covariant''' is a polynomial in ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>, ''x'', ''y'' that is invariant, so an invariant is a special case of a covariant where the variables ''x'' and ''y'' do not occur. More generally still, a '''simultaneous invariant''' is a polynomial in the coefficients of several different forms in ''x'' and ''y''.
In terms of representation theory, given any representation ''V'' of the group ''SL''<sub>2</sub>('''C''') one can ask for the ring of invariant polynomials on ''V''. Invariants of a binary form of degree ''n'' correspond to taking ''V'' to be the (''n'' + 1)-dimensional irreducible representation, and covariants correspond to taking ''V'' to be the sum of the irreducible representations of dimensions 2 and ''n'' + 1.
The invariants of a binary form are a graded algebra, and {{harvtxt|Gordan|1868}} proved that this algebra is finitely generated if the base field is the complex numbers.
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==Examples==
A form ''f'' is itself a covariant of degree 1 and order ''n''.
The [[discriminant]] of a form is an invariant.
The [[resultant]] of two forms is a simultaneous invariant of them.
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