Sure, let's address each part of the question regarding the algebraic expression [tex]\(x^3 y + x^2 y^2 - x y^3\)[/tex]:
### a) What is the type of the given expression?
The given expression [tex]\(x^3 y + x^2 y^2 - x y^3\)[/tex] is a Polynomial.
### b) Why is the given expression a polynomial?
The given expression is a polynomial because it consists of terms that are the product of constants and variables raised to non-negative integer powers. In a polynomial, you only find terms with variables having non-negative integer exponents, and this expression satisfies that condition.
### c) Find the degree of the given polynomial.
To find the degree of a polynomial, you need to determine the highest sum of the exponents of the variables in any term. Let's break down the degree calculation for each term:
1. First term: [tex]\(x^3 y\)[/tex]
- The exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are 3 and 1, respectively.
- The sum of the exponents is [tex]\(3 + 1 = 4\)[/tex].
2. Second term: [tex]\(x^2 y^2\)[/tex]
- The exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are 2 and 2, respectively.
- The sum of the exponents is [tex]\(2 + 2 = 4\)[/tex].
3. Third term: [tex]\(-x y^3\)[/tex]
- The exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are 1 and 3, respectively.
- The sum of the exponents is [tex]\(1 + 3 = 4\)[/tex].
The degrees of the individual terms are 4, 4, and 4. The highest of these sums is 4. Therefore, the degree of the polynomial [tex]\(x^3 y + x^2 y^2 - x y^3\)[/tex] is 4.
