Problem 38 - Elementary Algebra

Question

For the following problems, classify each of the polynomials as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term. $$ \left(2 x y^{3}\right)^{0}, \quad x y^{3} \neq 0 $$

Step-by-Step Solution

Answer

Based on the given polynomial, \((2xy^3)^0\), with the condition \(xy^3 \neq 0\), classify the polynomial, its degree, and the numerical coefficient of each term. Classification: Monomial Degree: 0 Numerical Coefficient: 1

Step 1: Simplify the term using the properties of exponents

We have the term \((2xy^3)^0\). According to the properties of exponents, any non-zero expression raised to the power of 0 is equal to 1. Therefore, we can simplify the term as: $$ (2xy^3)^0 = 1 $$

Step 2: Classify the polynomial

Since the simplified polynomial is a single term (1), it is a monomial. Monomials have only one term, while binomials have two, and trinomials have three terms.

Step 3: Determine the degree of the polynomial

The degree of a polynomial is the sum of the exponents of the variables in a term. Since our term is a constant (1), it does not have any variables. Therefore, the degree of the polynomial is 0.

Step 4: Write the numerical coefficient of each term

The numerical coefficient is the number that multiplies the variables in a term. Since our simplified term is the constant 1, the numerical coefficient is 1. So, the polynomial is a monomial with a degree of 0 and a numerical coefficient of 1.

Key Concepts

Monomial

A monomial is a type of polynomial that contains only a single term. This term can be a constant number, like 7, or involve variables, such as \( 3x^2 \) or \( -4xyz \). The main characteristics that define a monomial are:

• It contains only one term.
• The term may include a numerical coefficient and variables that can have exponents.
• It does not contain addition or subtraction signs within itself, unless it's \( +0 \) or \( -0 \).

For example, in the exercise you encountered, simplifying \((2xy^3)^0\) resulted in the value 1. 1 can be viewed as a monomial because it is a single term with no variables. Monomials are important because they are the building blocks of other types of polynomials, such as binomials (two terms) and trinomials (three terms). Understanding the concept of a monomial helps in simplifying and solving various algebraic expressions.

Degree of Polynomial

The degree of a polynomial is determined by the highest sum of exponents of the variables in any term of the polynomial. It gives us an idea of the polynomial's complexity. Here's what you need to remember:

• For a monomial like \(5x^7\), the degree is simply the exponent of the variable, which is 7.
• In a polynomial with multiple terms, you must find the term with the highest degree.
• Constants are special cases. They have a degree of 0 since no variables exist.

In the step-by-step solution, the simplified polynomial \(1\) was analyzed. Being a constant, it has no variables, so its degree is 0. Knowing how to find the degree of a polynomial helps when working on polynomial functions and graphing their behavior.

Numerical Coefficient

The numerical coefficient of a term within a polynomial is the number that multiplies the variable or variables. It is essential to grasp this concept as it signifies the "real number" part of any term:

• In a term like \(8x^3\), the numerical coefficient is 8.
• If there's no obvious number, it's usually understood to be 1, as in \( x^2 \) which can be seen as \(1x^2\).
• Negative signs can also be part of the coefficient. For \(-3xy\), the numerical coefficient is -3.

In the provided example, the simplified term was the constant \(1\). This makes the numerical coefficient 1. Recognizing numerical coefficients is helpful in operations like addition, subtraction, and multiplication of polynomials, as they help determine the value of the expression when variables are assigned a particular number.