Academic Notes on Special Polynomials: Complete, Ordered, and Homogeneous

Complete Polynomials

A complete polynomial (in Spanish, Polinomio completo) is a specific class of polynomial that contains every single power of a chosen variable, from its highest exponent (the degree of the polynomial) all the way down to the exponent of zero. The exponent of zero is fundamentally represented by the independent term, which is the constant in the expression.

A polynomial is verified as complete if, when observing the variable $x$, it possesses terms for every integer power starting from the maximum observed degree. For example, if a polynomial is of degree $3$, it must contain terms with $x^3$, $x^2$, $x^1$, and an independent term ($x^0$) to be considered complete.

Example provided: $F(x) = 3x^2 + 10 - x + 7x^3 + 2x^2$

In this specific case, the polynomial $F(x)$ is complete because it contains the following exponents for the variable $x$:

1. The third power ($3$), found in the term $7x^3$.

2. The second power ($2$), found in the terms $3x^2$ and $2x^2$.

3. The first power ($1$), found in the term $-x$ (which is implicitly $-1x^1$).

4. The zero power ($0$), found in the independent term $10$.

Ordered Polynomials

An ordered polynomial (in Spanish, Polinomio ordenado) is one in which the exponents of a specific variable are arranged in a particular numerical sequence. This sequence must be either strictly increasing (ascending) or strictly decreasing.

In an increasing ordered polynomial, the powers of the variable grow larger with each subsequent term. In a decreasing ordered polynomial, the powers of the variable diminish as you read across the expression from left to right.

Example provided: $R(x) = 8x^3 + 5x^2 - 2x + 3$

In the polynomial $R(x)$, the exponents for the variable $x$ are as follows:

• The first term ($8x^3$) has an exponent of $3$.

• The second term ($5x^2$) has an exponent of $2$.

• The third term ($-2x$) has an exponent of $1$.

• The fourth term ($3$) has an exponent of $0$.

Because the exponents follow the sequence $3, 2, 1, 0$, the polynomial $R(x)$ is classified as being ordered in a decreasing manner (forma decreciente) with respect to the variable $x$.

Homogeneous Polynomials

A homogeneous polynomial (in Spanish, Polinomio homogéneo) is defined by the equality of the absolute degrees ($GA$) of all its individual terms. The absolute degree of a term in a multi-variable polynomial is calculated by summing the exponents of all the variables within that specific term.

If every term in the polynomial results in the same numerical sum of exponents, the entire polynomial is said to be homogeneous. This constant value is known as the degree of homogeneity.

Example provided: $M(x; y) = 9x^2y^6 - \frac{2}{5}xy^7 + 4x^5y^3$

To determine if $M(x; y)$ is homogeneous, we calculate the absolute degree ($GA$) for each term:

1. For the term $9x^2y^6$, we sum the exponents of $x$ and $y$: $GA = 2 + 6 = 8$.

2. For the term $-\frac{2}{5}xy^7$, we sum the exponents: $GA = 1 + 7 = 8$.

3. For the term $4x^5y^3$, we sum the exponents: $GA = 5 + 3 = 8$.

Since the absolute degree for every term is consistently $8$, the polynomial is homogeneous with a degree of homogeneity equal to $8$.