How to Find the Degree of A Polynomial in Two Variables?

Polynomial can be of any degree and particularly if there is one variable in the polynomial, then it becomes easy to find the degree of polynomial. This is because in case of polynomial with just one variable has degree equals to the highest power of the variable in that expression. For instance, suppose we have a variable polynomial as: F (x, y) = 4 x⁷ + x⁶ + 4 x⁵ – 8 x² + 54 x + 7 x + 8. In this polynomial it is clear that degree is 7 as the expression 4 x⁷ represents the highest power of 'x'. Now how to find the degree of a polynomial in two variables?

In case polynomial contains two or more variables then we calculate the sum of powers of the variables to evaluate the degree. This can be explained with the help of an example: Suppose we have a polynomial of the form F (x, y) = 4 x⁴ y³ + x⁶ + 4 y⁵ – 8 x y² + 54 xy + 7x. In the given expression we can - not directly declare the degree of polynomial as 6 which is obtained by judging the expression x⁶ in the polynomial. In fact the sum of powers of x and y in all expression individually has to be evaluated to get the final degree of the polynomial. So, in the given polynomial we have the following summations of powers:

4 x⁴ y³ = 7,

x⁶ = 6,

4 y⁵ = 5,

8 x y² = 3,

54 x y = 2,

7 x = 1,

Thus we see that expression 4x⁴ y³ has the summation of powers maximum as compared to other expression. Thus we can finalize the degree of polynomial as 7.