Question
Determine if the expression minus, start fraction, x, cubed, divided by, 4, end fraction, plus, 4, y−
4
x
3
+4y is a polynomial or not. If it is a polynomial, state the type and degree of the polynomial.
The given expression ______ a polynomial. The polynomial is a _____
and has a degree of ______.
4
x
3
+4y is a polynomial or not. If it is a polynomial, state the type and degree of the polynomial.
The given expression ______ a polynomial. The polynomial is a _____
and has a degree of ______.
Answers
GPT-4o mini
To determine whether the expression \(-\frac{x^3}{4} + 4y\) is a polynomial, we can analyze its individual components.
1. **Polynomial Definition**: A polynomial is an expression formed by the sum of terms, where each term consists of a variable raised to a non-negative integer exponent and is multiplied by a coefficient (which can be a constant or a variable).
2. **Components of the Expression**:
- The term \(-\frac{x^3}{4}\) is a polynomial term because it consists of a variable \(x\) raised to the exponent of 3 (a non-negative integer) multiplied by the coefficient \(-\frac{1}{4}\).
- The term \(4y\) is also a polynomial term since \(y\) is a variable to the first power (also a non-negative integer) multiplied by the coefficient \(4\).
3. **Degree of the Polynomial**:
- The degree of a polynomial is the highest exponent of its variable in the expression. In this case, the highest exponent is \(3\) (from the term \(-\frac{x^3}{4}\)).
Since both terms are polynomial terms, and thus the overall expression is a sum of polynomial terms, we can conclude:
The given expression **is** a polynomial. The polynomial is a **two-variable polynomial** and has a degree of **3**.
1. **Polynomial Definition**: A polynomial is an expression formed by the sum of terms, where each term consists of a variable raised to a non-negative integer exponent and is multiplied by a coefficient (which can be a constant or a variable).
2. **Components of the Expression**:
- The term \(-\frac{x^3}{4}\) is a polynomial term because it consists of a variable \(x\) raised to the exponent of 3 (a non-negative integer) multiplied by the coefficient \(-\frac{1}{4}\).
- The term \(4y\) is also a polynomial term since \(y\) is a variable to the first power (also a non-negative integer) multiplied by the coefficient \(4\).
3. **Degree of the Polynomial**:
- The degree of a polynomial is the highest exponent of its variable in the expression. In this case, the highest exponent is \(3\) (from the term \(-\frac{x^3}{4}\)).
Since both terms are polynomial terms, and thus the overall expression is a sum of polynomial terms, we can conclude:
The given expression **is** a polynomial. The polynomial is a **two-variable polynomial** and has a degree of **3**.