Verified Answer :

  • The polynomial is \(10x^2 + 2x^m y - 4y\), with degrees 2, \(m+1\), and 1 respectively.
  • The degree of a polynomial is the maximum degree of its terms, so \(\max(2, m+1, 1) = 3\).
  • Solving for \(m+1=3\), we find \(m=2\).
  • The final answer is \(\boxed{2}\).

Explanation

  1. Analyze the Polynomial and Definition of Degree

The polynomial given is \(P(x, y) = 10x^2 + 2x^m y - 4y\). A polynomial consists of several terms. The degree of a term is defined as the sum of the exponents of its variables. For a polynomial in multiple variables, the degree of the polynomial is the maximum of the degrees of its individual terms. We are given that the degree of this polynomial is 3.

  1. Identify the Degrees of Individual Terms

We identify the degree of each term in the polynomial:

  1. The degree of the first term, \(10x^2\), is 2.
  2. The degree of the second term, \(2x^m y\), is \(m + 1\) (since the exponent of \(x\) is \(m\) and the exponent of \(y\) is 1).
  3. The degree of the third term, \(-4y\), is 1 (since the exponent of \(y\) is 1).

The degree of the polynomial \(P(x, y)\) is the maximum of these degrees, which is given by:
\[\max(2, m + 1, 1) = 3\]

  1. Solve for m

From the equation \(\max(2, m + 1, 1) = 3\), it follows that the maximum must be \(m + 1\), because 2 and 1 are both less than 3. Therefore, we set:
\[m + 1 = 3\]
Solving for \(m\):
\[m = 3 - 1 = 2\]
Thus, the value of \(m\) is 2. Substituting \(m=2\) into the term \(2x^m y\), we get \(2x^2 y\), which has a degree of \(2+1=3\), confirming our result.

Examples

In engineering and physics, polynomials are used to model the relationship between different physical quantities. For instance, if you are modeling the volume of a geometric shape like a box where the length, width, and height are functions of some variables, the expression for volume will be a polynomial. Understanding the degree of a polynomial allows scientists to determine the complexity and the behavior of such models. Finding unknown parameters like 'm' ensures that the mathematical model correctly matches the physical constraints of the system, such as the total dimension or scale of the object.