Question

Determine if the expression 7, y, minus, 6, y, squared, minus, y, to the power 57y−6y
2
−y
5
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 ______.

Answer 1

To determine whether the expression \( 7y - 6y^2 - y^{57} \) is a polynomial, we need to check the following criteria:

1. The expression should only contain non-negative integer powers of the variable \( y \).
2. Coefficients can be any real or complex numbers.

Let's analyze the given expression:
- \( 7y \) is a polynomial term with a degree of 1.
- \( -6y^2 \) is a polynomial term with a degree of 2.
- \( -y^{57} \) is a polynomial term with a degree of 57.

Since all terms are in the form of \( ay^n \) where \( n \) is a non-negative integer (and \( a \) is a coefficient), the entire expression satisfies the criteria for being a polynomial.

Now we can summarize:
- The given expression **is** a polynomial.
- The polynomial is a **trigonometric polynomial** (not used in strict polynomial definitions) since it consists solely of the powers of the variable \( y \).
- The degree of the polynomial is the highest degree of the terms, which is **57**.

Thus, the final answer is:
The given expression **is** a polynomial. The polynomial is a **trigonometric polynomial** and has a degree of **57**.