Polynomials in two variables

Let's take a look at polynomials in two variables.  Polynomials in two variables are algebraic expressions containing terms in the form axⁿ y ^m .  The degree of each term is the sum of the exponents in each term & the degree of the polynomial is the largest such sum in polynomial in two variables.

Following are some examples of polynomials in two variables and their degrees.

x² y - 6x³ y¹² + 10x² - 7 y + 1                                      degree : 15

6x⁴ + 8 y ⁴ - xy ²                                                                      degree : 4

x⁴ y ² - x³ y³ - xy + x⁴                                                degree : 6

6x¹⁴ -10 y³ + 3x -11y                                                  degree : 14

In these sort of polynomials not every term have to have both x's & y's in them, actually as we see in the last instance they don't have to have any terms which contain both x's and y's. Also, the degree of the polynomial might come from terms involving only one variable.  Note as well that multiple terms might have the same degree.

We also can talk about polynomials in three variables, or four variables or as several variables as we require.