What does biquadratic mean?
In mathematics, a biquadratic is a polynomial of the fourth degree, or an equation built from one. The term is most often used today for the special quartic form ax⁴ + bx² + c, which contains only even powers of the variable; such equations yield readily to the substitution y = x², which reduces them to ordinary quadratics. The word also serves as an adjective meaning 'relating to the fourth power', as in biquadratic residues, a concept in number theory concerned with remainders left when fourth powers are divided by a prime. Though 'quartic' has largely displaced 'biquadratic' as the everyday label for degree-four polynomials among modern mathematicians, the term persists in textbooks, historical treatments of algebra, and specialised contexts. Its name reflects its logic: because x⁴ is the square of x², a biquadratic problem is, quite literally, a quadratic applied twice.
A quartic polynomial, especially one of the form ax⁴ + bx² + c with no odd-powered terms; also, an equation involving such a polynomial.
"Ferrari's method handles any quartic, but a simple biquadratic needs only a substitution."
Often restricted to even-power quartics in modern usage.
Relating to the fourth power or to fourth-degree expressions.
"Gauss developed a theory of biquadratic residues in his work on higher reciprocity."
Standard English plural, used when referring to multiple fourth-degree polynomials or equations.
"The textbook's final chapter works through several biquadratics."
Despite containing 'bi-', a biquadratic has nothing to do with bicycles or pairs — the prefix refers to squaring twice, since x⁴ is just (x²)².
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of biquadratic
Biquadratic derives from Latin roots: 'bis', meaning twice, combined with 'quadratus', meaning squared, itself from 'quadrare' (to make square) and 'quadrus' (a square). The term entered English mathematical usage in the early modern period, alongside related forms such as 'quadratic' and 'quadrate'. Its cognates include 'quadruple', 'quadrature', and 'quadratic', all descending from the same Latin family concerning squares and foursomes. The prefix does not indicate two terms but rather the act of squaring twice.
Related word forms
How biquadratic is actually used
Biquadratic belongs chiefly to mathematical register. In modern usage, mathematicians more often say 'quartic' for general fourth-degree polynomials, reserving 'biquadratic' for the special case where odd powers are absent (ax⁴ + bx² + c). The term also appears in number theory for 'biquadratic residues' (fourth-power residues) and in older literature on algebraic equations.
Easily confused with biquadratic
A quadratic involves the second power (x²), while a biquadratic involves the fourth power (x⁴); despite the name, 'bi-' here signals two successive squarings rather than two terms.