Homogeneous Polynomials

A polynomial in standard form is called a homogeneous polynomial if all of its terms have the same total degree.

A Practical Example

Consider the following polynomial:

$$ ab^3 + a^2bc + a^2b^2 $$

This expression is the algebraic sum of three monomials.

Each monomial has a total degree of 4. Therefore, the polynomial is homogeneous.

Example 2

This polynomial is not homogeneous because the first three terms are of degree 3, while the constant term (5) has degree zero:

$$ ab^2 + abc + a^2b + 5 $$

Note: Constants are always considered degree-zero terms because they can be written as the product of the number and any variable raised to the zero power. For example: $$ 5 = 5 \cdot 1 = 5 \cdot a^0 $$

Example 3

This polynomial is also not homogeneous: the first two terms have degree 3, but the last term has degree 2:

$$ ab^2 + abc + ab $$

And so on.