Problem 29 - Elementary Algebra

Question

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term. $$ 5 a^{3} b $$

Step-by-Step Solution

Answer

Answer: The polynomial \(5a^3b\) is a monomial with a degree of 3 and a numerical coefficient of 5.

Step 1: Classify the polynomial as a monomial, binomial, or trinomial

The given polynomial is \(5a^3b\). Since it has only one term, it is classified as a monomial.

Step 2: Find the degree of the polynomial

The degree of a polynomial is the highest power of the variable(s) in its terms. In the given polynomial, the highest power is 3 (from \(a^3\)); thus, the degree of the polynomial is 3.

Step 3: Write down the numerical coefficient of the term

In the given polynomial, the numerical coefficient of the term \(5a^3b\) is 5.

Key Concepts

Monomial

A monomial is a type of polynomial that has only one term. In mathematical terms, it is an expression consisting of numbers, variables, or combinations of both that are multiplied together.
For example, in the polynomial \(5a^3b\), there is only one term in the expression. Hence, it is classified as a monomial. Monomials are simple to work with because they do not involve addition or subtraction between different terms.
Key characteristics of a monomial include:

• Having only one term.
• Being composed of variables, numbers, or both in multiplication.
• Having a whole number as the exponent of any variable.

Understanding monomials is foundational to comprehending more complex polynomials and helps in learning operations such as multiplication and division involving polynomials.

Degree of a Polynomial

The degree of a polynomial is one of its most important characteristics. It helps determine the polynomial's behavior and is essential for understanding its graph. The degree is found by looking at the term with the highest sum of exponents of its variables.
In the given monomial \(5a^3b\), to determine the degree, we sum the exponents of the variables present in the term. Here, the exponent of \(a\) is 3, and the exponent of \(b\) is 1 (when explicitly written as \(b^1\)). Thus, the degree of the polynomial is \(3 + 1 = 4\).
Important points about the degree of a polynomial:

• The degree is a whole number.
• It indicates the highest power of the variables in the polynomial.
• In simple terms, the degree tells us the most you can combine the variables using multiplication.

The degree of a polynomial plays a critical role in simplifying polynomials and solving equations involving them. It also affects the shape of the polynomial's graph.

Numerical Coefficient

A numerical coefficient is a number that stands in front of a variable in a term and acts as its multiplier. It quantifies how many times the term should be considered in an expression.
For example, in the monomial \(5a^3b\), the numerical coefficient is 5. This indicates that \(a^3b\) is being multiplied by 5.
Points to remember about numerical coefficients:

• They can be positive, negative, or zero.
• They can be whole numbers, fractions, or decimals.
• The absence of a visible coefficient implies a numerical coefficient of 1.

The numerical coefficient is essential when performing operations like addition and subtraction with polynomials, ensuring each term is effectively accounted for and accurately combined.