Problem 38 - Elementary and Intermediate Algebra

Question

Find the degree of each polynomial. See Example \(1 .\) $$ \frac{1}{2} y^{3}+4 y^{2} $$

Step-by-Step Solution

Answer

The degree of the polynomial is 3.

Step 1: Identify each term of the polynomial

The given polynomial is \( \frac{1}{2} y^{3} + 4y^{2} \). It consists of two terms: \( \frac{1}{2} y^{3} \) and \( 4y^{2} \).

Step 2: Find the degree of each term

The degree of a term is determined by the exponent attached to the variable. For \( \frac{1}{2} y^{3} \), the degree is 3 because \( y \) is raised to the power of 3. For \( 4y^{2} \), the degree is 2 because \( y \) is raised to the power of 2.

Step 3: Determine the degree of the polynomial

The degree of a polynomial is the highest degree of any term within the polynomial. Compare the degrees of the individual terms: 3 for \( \frac{1}{2} y^{3} \) and 2 for \( 4y^{2} \). The highest degree is 3.

Key Concepts

Polynomial Terms

In mathematics, a polynomial is made up of one or more terms, each known as a polynomial term. A term is essentially a combination of constants and variables. For example, in the polynomial \( \frac{1}{2} y^{3} + 4y^{2} \), there are two polynomial terms: \( \frac{1}{2} y^{3} \) and \( 4y^{2} \). Each term can have:

• A numerical coefficient (like \( \frac{1}{2} \) or \(4\)).
• One or more variables (such as \( y \) in this case).
• An exponent, which tells you how many times the variable is multiplied by itself (like \( y^{3} \) or \( y^{2} \)).

Understanding the components of polynomial terms is crucial because it helps in analyzing and simplifying polynomial expressions. Knowing the terms also helps identify the degree of each term, an essential step in solving polynomial-related problems.

Polynomial Expression

A polynomial expression is a mathematical phrase that can contain multiple terms. These terms are combined by addition and subtraction. A polynomial expression needs to follow specific rules. It can only include operations of addition, subtraction, and multiplication combined with variables that have non-negative integer exponents. An expression like \( \frac{1}{2} y^{3} + 4y^{2} \) is a classic example of a polynomial. Poly means "many," and nomial means "terms," reflecting its structure:

• It can have constants, like \( \frac{1}{2} \) and \( 4 \).
• It can include terms with variables raised to positive integer powers, like \( y^{3} \) and \( y^{2} \).

The goal when working with polynomial expressions is often to simplify them or to find certain properties, such as the degree, which offers insights into the polynomial's behavior and complexity.

Degree of a Term

The degree of a term is a key concept in understanding polynomials. It is determined by the exponent of the variable within the term. For instance, if you see \( y^3 \), the degree is 3, because the variable \( y \) has an exponent of 3. This is crucial in identifying the highest degree term in a polynomial.Whether a term has a single variable or multiple variables, the degree applies to the combined exponents of all variables in that term. For a single-variable term like \( 4y^{2} \), the degree is simply the exponent of \( y \), which is 2. In more complex terms, where multiple variables exist, you would add the exponents of the variables to find the term's degree.Knowing each term's degree allows you to determine the degree of the entire polynomial, which is the largest degree of any individual term. This is why, in our given example, the polynomial \( \frac{1}{2} y^{3} + 4y^{2} \) has a degree of 3, as the highest degree term within it is \( \frac{1}{2} y^{3} \). Understanding the degree helps to predict the polynomial's graph behavior and solve complex equations.