Problem 24 - 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. $$ 9 g $$

Step-by-Step Solution

Answer

Question: Classify the polynomial \(9g\) and state its degree and numerical coefficient. Answer: The polynomial \(9g\) is a monomial with a degree of 1 and a numerical coefficient of 9.

Step 1: Classify the polynomial

The given polynomial is \(9g\). It has only one term, which is g multiplied by a constant 9. Since it has only one term, it is a monomial.

Step 2: Determine the degree of the polynomial

The degree of the polynomial is the highest exponent of its variable. In this case, the variable is g and it is raised to the power of 1 (since there is no exponent written, it is assumed to be 1). So, the degree of the polynomial is 1.

Step 3: Write the numerical coefficient of each term

The numerical coefficient of the term in the given polynomial is the number that is multiplied by the variable. In this case, the number that is multiplied by g is 9. Therefore, the numerical coefficient of the term is 9. To summarize, the given polynomial, \(9g\) is a monomial with a degree of 1 and has a numerical coefficient of 9.

Key Concepts

Monomials

Monomials are the simplest form of polynomials, consisting of just one term. They can include numbers, variables, or a combination of both:

• 9g
• 4x^2
• -3a

In each of these examples, there is only one term making them monomials. Monomials are important because they serve as the building blocks for more complex polynomials, like binomials and trinomials. When identifying a monomial, look for a single term that may include a variable raised to a power, but is not separated by addition or subtraction. These can include expressions like 5y and constants like 7.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable present in the polynomial. This is a crucial indicator as it affects the graph's shape and the polynomial's behavior. To find the degree:

• Identify the variable's highest exponent in the polynomial.
• If there is no visible exponent, it is implicitly understood to be 1. For example, in the polynomial 7x, the degree is 1 because x is actually x to the power of 1.
• For monomials like 9g, the degree is 1, since there are no other terms and the variable g does not appear with an exponent.

Understanding the degree helps in grasping how a polynomial behaves in an equation or graph.

Numerical Coefficient

A numerical coefficient is the mathematical constant that multiplies the variable in a polynomial term. It gives weight or intensity to the variable, representing how many times the variable is being multiplied.

• In the monomial 9g, the coefficient is 9.
• With 4x^2, the coefficient is 4.
• If no number is present, like in the variable x, then the numerical coefficient is 1 because x equals 1x.

The numerical coefficient is always placed directly in front of the variable. It's important for solving equations and understanding the term's contribution within the polynomial. Remember that the coefficient can be a positive, negative, whole number, or fractional value.