Question

Determine if the expression 7, x, cubed, y, squared, plus, 2, y, to the power 47x
3
y
2
+2y
4
is a polynomial or not. If it is a polynomial, state the type and degree of the polynomial.

Answer 1

To determine if the expression \( 7x^3y^2 + 2y^{47} \) is a polynomial, we examine the structure of the terms involved.

A polynomial is defined as an expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and constant coefficients.

Now let's break down the given expression:

1. **First term**: \( 7x^3y^2 \)
- Here, \( x \) is raised to the power of 3 (a non-negative integer), and \( y \) is raised to the power of 2 (also a non-negative integer). The coefficient is 7, which is a constant. Hence, this term is valid in a polynomial.

2. **Second term**: \( 2y^{47} \)
- In this term, \( y \) is raised to the power of 47 (a non-negative integer), and the coefficient of this term is 2. This term is also valid in a polynomial.

Since both terms consist of variables raised to non-negative integer powers and are combined using addition, the entire expression \( 7x^3y^2 + 2y^{47} \) is a polynomial.

### Now, let's state the type and degree of the polynomial:

1. **Type of Polynomial**:
This is a **multivariable polynomial** (in \( x \) and \( y \)).

2. **Degree of the Polynomial**:
The degree of a polynomial is determined by the highest sum of the exponents of the variables in each term.

- For the term \( 7x^3y^2 \), the sum of the exponents is \( 3 + 2 = 5 \).
- For the term \( 2y^{47} \), the sum of the exponents is \( 47 \).

The highest degree among the terms is 47.

### Conclusion:
The expression \( 7x^3y^2 + 2y^{47} \) is a polynomial, and it is a multivariable polynomial of degree **47**.