To determine if the expression [tex]\(-\frac{y^5}{4} - 3y^3 - y^4\)[/tex] is a polynomial, and if so, to find its type and degree, let's analyze the expression step-by-step.
1. Identify the terms in the expression:
The given expression is:
[tex]\[
-\frac{y^5}{4} - 3y^3 - y^4
\][/tex]
2. Confirm if the expression is a polynomial:
A polynomial is an algebraic expression made up of terms which are summed or subtracted together, where each term includes a variable raised to a non-negative integer power and has a coefficient. In simpler terms, each term should have the form [tex]\(a y^n\)[/tex] where [tex]\(a\)[/tex] is a real number and [tex]\(n\)[/tex] is a non-negative integer.
Let's examine each term given:
- The term [tex]\(-\frac{y^5}{4}\)[/tex] is [tex]\(-\frac{1}{4} y^5\)[/tex], which has a power of [tex]\(5\)[/tex], and [tex]\(\frac{1}{4}\)[/tex] is a real number.
- The term [tex]\(-3y^3\)[/tex] has a power of [tex]\(3\)[/tex], and [tex]\(-3\)[/tex] is a real number.
- The term [tex]\(-y^4\)[/tex] is [tex]\(-1 \cdot y^4\)[/tex] with a power of [tex]\(4\)[/tex] and [tex]\(-1\)[/tex] is a real number.
All terms fit the form of a polynomial.
3. Degree of the polynomial:
The degree of a polynomial is the highest power of the variable within the expression. From the given terms, the highest power of [tex]\(y\)[/tex] is [tex]\(5\)[/tex].
Therefore, the degree of this polynomial is [tex]\(5\)[/tex].
4. Type of polynomial:
The type of polynomial based on its terms:
- Based on the highest degree term (-[tex]\(\frac{1}{4} y^5\)[/tex]), it is a fifth-degree polynomial.
- The expression is not sparse because it doesn't have gaps between the degrees (each step down in power only skips degree [tex]\(2\)[/tex], which is common and does not necessarily denote sparsity in this context).
Therefore,
- The given expression [tex]\(-\frac{y^5}{4} - 3y^3 - y^4\)[/tex] is a polynomial.
- The degree of the polynomial is [tex]\(5\)[/tex].
- The type of the polynomial is a standard polynomial (if additional specificity is required beyond standard polynomial types such as monomial, binomial, or trinomial, it might loosely fit a sparse polynomial description due to the missing [tex]\(y^2\)[/tex] term, but typically sparse implies larger gaps between significant terms).
Thus, summarizing:
[tex]\[
\text{The expression is a polynomial, its degree is 5, and it is a sparse polynomial.}
\][/tex]
