Answer:
the degree of homogeneity is 20.
Step-by-step explanation:
In a polynomial P (x, y), homogeneous and complete in "x" and "y", the sum of the absolute degrees of all its terms is 420. What is its degree of homogeneity?
A homogeneous polynomial is one in which all monomials have the same degree.
This is an example of a homogeneous polynomial of degree 4 (the degree of all monomials is 4):
x ^ 4 + 3x ^ 3y + 2x ^ 2y ^ 2 + xy ^ 3 + 8y ^ 4.
As you can see, the sum of the exponents of the variables x, y in each monomial is 4.
And the number of terms is 5, that is, it is the degree of homogeneity plus 1.
In relation to the sum of the absolute degrees of all monomials or terms it will be: 4 + 4 + 4 + 4 + 4 = 4 * 5 = 20.
In general, you can say that the sum of the absolute degrees in a homogeneous polynomial will be the degree of each monomial by the number of terms = degree * (degree + 1)
Calling n, the degree of our polynomial, it must be fulfilled:
n (n + 1) = 420
=> n ^ 2 + n = 420
=> n ^ 2 + n - 420 = 0
Factoring:
(n + 21) (n - 20) = 0
=> n = -21 and n = 20.
Only the positive value makes sense, therefore n = 20.
In other words, the polynomial is of the form (excluding the coefficients):
x ^ 20 + x ^ 19 y + x ^ 18 y ^ 2 + x ^ 17 y ^ 3 + .... x ^ 3 y ^ 17 + x ^ 2y ^ 18 + xy ^ 19 + y ^ 20
That polynomial has 21 terms.
So the sum of the degrees will be 20 * 21 = 420, as required in the statement.
Therefore, the degree of homogeneity is 20.
