Certainly! Let's carefully address both parts of your question.
### Step 1: Determining if the Expression is a Polynomial
A polynomial is an expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer powers of the variables.
Let's look at each term in the expression \(x^2 + y^2 + z^2 - 3xyz\):
- \(x^2\): This term involves the variable \(x\) raised to the power of 2.
- \(y^2\): This term involves the variable \(y\) raised to the power of 2.
- \(z^2\): This term involves the variable \(z\) raised to the power of 2.
- \(-3xyz\): This term involves the variables \(x\), \(y\), and \(z\) each raised to the power of 1, and they are multiplied together.
All of these terms involve variables raised to non-negative integer powers and involve addition, subtraction, and multiplication. Therefore, the given expression \(x^2 + y^2 + z^2 - 3xyz\) is indeed a polynomial.
### Step 2: Finding the Degree of the Polynomial
The degree of a polynomial is the highest degree of its monomials (individual terms) with non-zero coefficients. The degree of a monomial is the sum of the exponents of the variables in that term.
Let’s analyze the degree of each term in the polynomial:
- For \(x^2\): The degree is 2 (since \(x\) is raised to the power of 2).
- For \(y^2\): The degree is 2 (since \(y\) is raised to the power of 2).
- For \(z^2\): The degree is 2 (since \(z\) is raised to the power of 2).
- For \(-3xyz\): The degree is \(1 + 1 + 1 = 3\) (since each variable \(x\), \(y\), and \(z\) is raised to the power of 1, and they are multiplied together).
The highest degree among these terms is the degree of the term \(-3xyz\), which is 3.
### Final Answer
- The expression \(x^2 + y^2 + z^2 - 3xyz\) is a polynomial.
- The degree of this polynomial is 3.
