Reducible Polynomial

Summary: Two classifications of multivariate polynomials, stated in §94–§95. The order of a polynomial is the greatest degree of any single term (relevant to the study of algebraic curves). A polynomial is reducible if it is a product of two or more non-irrational factors, irreducible otherwise. Every homogeneous bivariate polynomial is reducible; irreducibility is checked by examining divisors.

Sources: chapter5

Last updated: 2026-04-23

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Order of a polynomial (§94)

The order of a polynomial is the greatest degree of any single term (source: chapter5, §94). Equivalently: the highest total degree that appears, ignoring which lower-degree terms may or may not be present.

Examples:

• $z^2+y^2+z^2+ay-a^2$ has order 2 (the $z^2,y^2$ terms are degree 2; the $ay$ is degree 1; the $-a^2$ is degree 0).
• $y^4+y{z}^3-a{y}^{2}x+abyz-a^2{y}^2+b^4$ has order 4.

Order is the classification used in the geometry of algebraic curves — a “second-order curve” is a conic, a “third-order curve” is a cubic, and so on. Order differs from degree only when the polynomial is heterogeneous: a homogeneous polynomial has a single degree, which is also its order.

Reducibility (§95)

A polynomial is reducible if it can be written as a product of two or more non-irrational (i.e. polynomial or rational) factors; otherwise irreducible (source: chapter5, §95).

Example of reducibility.

$y^4-z^4+2a{z}^3-2by{z}^2-a^2{z}^2+2abzy-b^2{y}^2=(y^2+z^2-az+by)(y^2-z^2+az-by).$

Every homogeneous bivariate polynomial is reducible. This follows from §91: any homogeneous polynomial of degree $n$ in $y,z$ is a product of $n$ linear factors $\alpha y+\beta z$ (see homogeneous-function).

Example of irreducibility. $y^2+z^2-a^2$ has no non-irrational factorization — it is irreducible (source: chapter5, §95).

Euler’s method for deciding reducibility is to “consider divisors” — i.e. test whether the polynomial has a polynomial factor of lower degree. No systematic algorithm is developed here.

Related pages

• homogeneous-function
• order-of-a-polynomial
• factoring-polynomials
• fundamental-theorem-of-algebra
• chapter-5-on-functions-of-two-or-more-variables