Problem 10 - College Algebra

Question

For the following exercises, identify the degree of the polynomial. $$ 6 y^{4}-y^{5}+3 y-4 $$

Step-by-Step Solution

Answer

The polynomial degree is 5.

Step 1: Identify each term in the polynomial

The polynomial given is \(6y^4 - y^5 + 3y - 4\). Identify the individual terms: \(6y^4\), \(-y^5\), \(3y\), and \(-4\).

Step 2: Determine the degree of each term

The degree of a term is the sum of the exponents of the variables in the term. For the given terms: - \(6y^4\) has a degree of 4. - \(-y^5\) has a degree of 5. - \(3y\) has a degree of 1. - \(-4\) (a constant) has a degree of 0.

Step 3: Identify the polynomial's degree

The degree of the polynomial is the highest degree of any term within the polynomial. Among the terms \(6y^4\), \(-y^5\), \(3y\), and \(-4\), the term \(-y^5\) has the highest degree, which is 5.

Key Concepts

Polynomial terms

In any polynomial, each expression separated by a plus or minus sign is called a term. Terms are like small parts that combine to form the bigger structure known as the polynomial. In the polynomial given in our exercise, which is \(6y^4 - y^5 + 3y - 4\), there are four terms:

• \(6y^4\)
• \(-y^5\)
• \(3y\)
• \(-4\)

Each term contains numbers and/or variables, and understanding how to identify these terms is crucial in solving polynomial problems.

Exponents

An exponent is a small number written to the upper right of a base number or variable, indicating how many times the base is multiplied by itself. For example, in the term \(y^4\), "4" is the exponent, and it means the variable \(y\) is multiplied by itself four times (\(y \times y \times y \times y\)).
Exponents are found in every term of a polynomial that includes a variable raised to a power. The role of exponents is significant when determining the degree of the polynomial and each individual term.

• In \(6y^4\), the exponent is 4.
• In \(-y^5\), the exponent is 5.
• In \(3y\), if not explicitly stated, the exponent is 1, since it's \(3y^1\).
• In \(-4\), as there is no variable, the exponent is considered 0.

Understanding exponents makes it easier to calculate the degree of terms.

Degree of a term

The degree of a term is determined by the sum of the exponents of its variables. If a term has multiple variables, you simply add up their exponents. For single-variable terms, the degree is simply the exponent itself. The degree gives us significant insight into the behavior and properties of a polynomial.

Degrees of the Terms

Let's look at each term:

• \(6y^4\) has a degree of 4 because the exponent of \(y\) is 4.
• \(-y^5\) has a degree of 5, due to the exponent of \(y\) being 5.
• \(3y\) has a degree of 1, as it's equivalent to \(3y^1\).
• \(-4\) is a constant term (no variables), and constants have a degree of 0.

To find the degree of a polynomial itself, identify the highest degree among its terms; here, it is 5 from the term \(-y^5\). Recognizing these degrees is vital in analyzing polynomials for both practical applications and theoretical mathematics.