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I used the term "order" in place of "degree" to define a polynomial. Are the terms "degree" and "order" of a polynomial the same in algebra?

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3 Answers

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From Wikipedia:

The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts

These "other concepts", however, are more advanced properties of a polynomial. If the polynomial is considered as a power series, for example, "order" means the non-zero coefficient of lowest degree. If the polynomial describes a spline, its order is conventionally the degree $+1$, referring to the number of knots needed to specify it.

However, unless there is the possibility of confusion with any of these other concepts, using "order" in place of "degree" should be fine.

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For polynomials the degree is more common than the order but there is no confusion if you use the word order instead.

For differential equations the order is commonly used instead of degree.

We say this is a third degree polynomial and that is a third order differential equation.

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In algebra, "degree" is sometimes called "order." The degree is the largest exponent of that variable (also called indeterminants):

$4x$ the degree is $1$ ( a variable without an exponent actually has an exponent of $1$),

$4x^3-x+3$ the degree is $3$ (the largest exponent of $x$),

$x^2+2x^5-x$ the degree is $5$ (the largest exponent of $x$),

$z^2-z+3$ the degree is $2$ (the largest exponent of z).

When a polynomial has more than one variable, we need to look at each term. Terms are separated by $+$ or $-$ signs. For each term: Find the degree by adding the exponents of each variable in each term, the largest such degree is the degree of the polynomial. What is the degree of this polynomial, $5xy^2-3x+5y^3-3$? The largest degree of those is $3$ (in fact two terms have a degree of $3$), so the polynomial has a degree of $3$. What is the degree of this polynomial, $4z^3+5y^2z^2+2yz$? The largest degree is $4$.

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