Degree of a Polynomial

The degree of a polynomial in standard (simplified) form is the highest degree among its terms.

A Practical Example

This polynomial consists of three terms. The first term, \( ab^3 \), has degree 4; the second, \( a^2b \), has degree 3; and the third, \( ab \), has degree 2.

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

Therefore, the degree of the polynomial is 4.

Note: The degree of a polynomial should be determined only after it has been simplified into standard form - that is, when all like terms have been combined. If the polynomial contains like terms, it must first be reduced. For example, the following expression is not of degree 3 because it’s not yet simplified: $$ ab^2 + 2ab - ab^2 + c $$ Let’s simplify it: $$ (ab^2 - ab^2) + 2ab + c $$ $$ 2ab + c $$ Now that the polynomial is in standard form, its degree is 2.

Degree of a Polynomial with Respect to a Variable

The degree of a polynomial with respect to a specific variable is the highest exponent of that variable among all the terms.

Example

Consider the following polynomial:

$$ ab^3 + a^2bc + ab + 5 $$

We can now determine the degree of the polynomial with respect to each variable:

• The degree with respect to a is 2, since that’s the highest exponent of a.
• The degree with respect to b is 3, which is the highest power of b in any term.
• The degree with respect to c is 1, as c appears only with an exponent of 1.

Note: The constant term (a number with no variables) is considered to have degree zero, because it can be written as the product of that number and a variable raised to the zero power. For example: $$ 5 = 5 \cdot a^0 $$

And so on.