Degree of a Monomial
The degree of a monomial is the sum of the exponents of all the variables it contains.
Only the exponents of the variables are considered - any numerical coefficients are ignored.
So, if a monomial contains no variables, its degree is zero.
Example. The monomial a2b3 has degree 5, since the exponents add up: 2 + 3 = 5. $$ a^2b^3 \ \ \text{monomial of degree 5} $$
The exponent of an individual variable in a monomial written in standard form is called the degree with respect to that variable.
Example. In the monomial a2b3, the degree with respect to the variable "a" is 2, since the exponent of "a" is 2. $$ a^2b^3 \ \ \text{degree 2 with respect to a} $$ Likewise, the degree with respect to "b" is 3, because the exponent of "b" is 3. $$ a^2b^3 \ \ \text{degree 3 with respect to b} $$
What if a variable has an exponent of 1?
When a variable has an exponent of 1, the exponent is typically not written, as it’s understood by convention. However, it still counts toward the total degree of the monomial.
$$ a = a^1 $$
$$ b = b^1 $$
$$ \vdots $$
Example. The monomial a2b has degree 3, since the exponent of "a" is 2 and the (implied) exponent of "b" is 1. So, 2 + 1 = 3. $$ a^2b \ \ \text{monomial of degree 3} $$ Here, the degree with respect to "a" is 2, and the degree with respect to "b" is 1.
When is the degree of a monomial zero?
A monomial has degree zero when it consists solely of a constant - i.e., a number with no variables.
This makes sense, because any variable raised to the power of zero equals 1.
For instance, the monomial 2 can be written as 2a0, since a0 = 1.
$$ 2 = 2 \cdot 1 = 2 \cdot a^0 $$
The same reasoning applies to any other nonzero constant.
Here are a few examples of monomials with degree zero: $$ 2 \ \ \text{monomial of degree 0} $$ $$ 3 \ \ \text{monomial of degree 0} $$ $$ 4 \ \ \text{monomial of degree 0} $$ $$ \vdots $$
When does a monomial have no degree?
In the special case of the zero monomial, no degree is assigned because its degree is considered undefined.
So, the numerical monomial 0 is not a monomial of degree zero - it simply has no degree at all.
$$ 0 \ \ \text{monomial with no defined degree} $$
This is because zero can result from multiplying variables raised to powers of any degree, making it impossible to assign a specific degree to the monomial.
$$ 0 = 0 \cdot x^0 = 0 \cdot x^1 = 0 \cdot x^2 = \dots $$
By contrast, constant (non-zero) numerical monomials such as 1, 2, 3, and so on all have degree zero.
This is because any non-zero number can be written as the product of itself and a variable raised to the zero power, which equals 1.
$$ 1 = 1 \cdot x^0 = 1 \cdot 1 $$
$$ 2 = 2 \cdot x^0 = 2 \cdot 1 $$
$$ 3 = 3 \cdot x^0 = 3 \cdot 1 $$
And so forth.
