5x4y2z

The term 5x4y2z denotes a monomial in three variables x, y, and z. In standard algebraic notation it is written as 5·x⁴·y²·z, with the coefficient 5 and exponents 4, 2, and 1 on the variables x, y, and z, respectively. The total degree of the monomial is the sum of all exponents, giving 4 + 2 + 1 = 7. The partial degree in each variable is the exponent of that variable: 4 in x, 2 in y, and 1 in z.

Within the polynomial ring ℤ[x, y, z] or ℚ[x, y, z], this monomial is an irreducible element because it cannot be expressed as a product of non‑unit polynomials of lower degree. Its presence in a polynomial equation or expression contributes a single term of total degree seven unless it is combined with other terms of the same total degree in a homogeneous polynomial.

In calculus, the partial derivatives of the monomial are straightforward: ∂/∂x (5x⁴y²z) = 20x³y²z, ∂/∂y = 10x⁴y z, and ∂/∂z = 5x⁴y². These derivatives illustrate how the growth rate of the function changes with each variable.

The monomial appears in multivariate expansions such as the binomial theorem generalized to several variables, multinomial expansions, and in the study of polynomial ideals. In algebraic geometry, a monomial like 5x⁴y²z is associated with a coordinate–wise scaling of a point in affine 3‑space; its vanishing set is the union of the coordinate hyperplanes x = 0, y = 0, or z = 0. Because its coefficient is non‑zero, it is never identically zero on any non‑empty open subset of ℝ³ or ℂ³. This basic building block underlies many constructions in commutative algebra and invariant theory.