Complete Polynomial
A polynomial is said to be complete with respect to a given variable if it includes all powers of that variable from the highest degree down to zero.
For example, the following polynomial is complete with respect to the variable \( x \) because it contains every power of \( x \) from degree 3 down to degree 0: \( x^3, x^2, x^1 \), and \( x^0 \).
$$ 4x^3 + 3x^2 - x + 3 $$
Note: The constant term in a polynomial can be interpreted as the coefficient of a variable raised to the zero power: $$ 3 = 3 \cdot 1 = 3 \cdot x^0 $$ So, the polynomial $$ 4x^3 + 3x^2 - x + 3 $$ can be rewritten in a more explicit - though redundant - form: $$ 4x^3 + 3x^2 - x^1 + 3x^0 $$ This version makes it easier to see that all powers of \( x \), from the highest to the lowest, are present.
If the constant term is missing, it can be assumed to be zero.
For example, this polynomial has no constant term:
$$ x^4 + 2x^3 - x^2 + 3x $$
In this case, the degree-zero term is absent.
However, we can express it in an equivalent form by adding a zero constant $ 0 = 0 \cdot x^0 $.
$$ x^4 + 2x^3 - x^2 + 3x + 0 \cdot x^0 $$
This shows that the polynomial is still complete with respect to \( x \).
Note: It's important to note that there's no universally accepted definition of a "complete polynomial" - the term can mean different things depending on the author. In many textbooks, a "complete polynomial" is understood as an expression that includes every power of the variable from the highest degree down to x0, even if some of the coefficients are zero. Under this interpretation, the polynomial $ x^4 + 2x^3 - x^2 + 3x $ is considered complete. Other authors, however, define a complete polynomial as one that includes all degrees of the variable with nonzero coefficients. According to this stricter definition, the same polynomial $ x^4 + 2x^3 - x^2 + 3x $ would not be considered complete, since the constant term is missing. In my own work, I follow the first definition.
A complete polynomial of degree \( n \) must contain at least \( n + 1 \) terms.
For instance, the following polynomial is of degree 4 and has five terms:
$$ x^4 + 2x^3 - x^2 + 3x + 0 $$
A polynomial may have more than \( n + 1 \) terms if it includes other variables besides \( x \).
For example, this polynomial is of degree 4 and is complete with respect to \( x \), but it contains six terms because it also includes terms in \( y \) that are not tied to \( x \):
$$ x^4y^3 + 2x^3y^2 - x^2 + 3x + 2y + 0 $$
As previously noted, the order of the terms does not affect whether a polynomial is considered complete.
For example, this polynomial is both complete and ordered:
$$ x^4 + 2x^3 - x^2 + 3x + 3 $$
While this one is complete but not ordered:
$$ 3x + x^4 - x^2 + 3 + 2x^3 $$
In both cases, the polynomial is complete with respect to \( x \).
And so on.
