Problem 25 - Algebra 1: Concepts and Skills

Question

Write the polynomial in standard form. Then identify the polynomial by degree and by the number of terms. $$ 20 m^{3} $$

Step-by-Step Solution

Answer

The polynomial \(20 m^{3}\) is already in standard form. Its degree is 3, and it is a monomial (one term).

Step 1: Write the Polynomial in Standard Form

The given polynomial \(20 m^{3}\) is already in standard form. The standard form for polynomials arranges the terms by degree in descending order. In this case, there is only one term, so no rearrangement is necessary.

Step 2: Identify the Polynomial by Degree

The degree of a polynomial is the highest power of the variable. In the given polynomial \(20 m^{3}\), the highest power of m is 3, so the degree of this polynomial is 3.

Step 3: Identify the Polynomial by the Number of Terms

A polynomial can be a monomial (one term), binomial (two terms), trinomial (three terms), or have four or more terms. The given polynomial \(20 m^{3}\) only has one term, so it is a monomial.

Key Concepts

Monomial

A monomial is a type of polynomial with only one term. Unlike other polynomials that may have multiple terms, a monomial consists solely of a constant multiplied by a variable raised to a non-negative integer power. An example of a monomial is \(20m^3\). Here, **20** is the constant, **m** is the variable, and the exponent **3** indicates the power to which the variable is raised.

• Simple to identify: One term only.
• Includes a constant and/or a variable.
• No addition or subtraction with other terms.

Recognizing monomials is important because they form the building blocks of more complex polynomials.

Degree of Polynomial

The degree of a polynomial is one of its most important characteristics. It represents the highest power of the variable present in the polynomial. For instance, in the monomial \(20m^3\), the degree is 3 because the variable \(m\) is raised to the power of 3.
Understanding the degree is crucial because it provides insight into the behavior and the shape of the polynomial's graph. Here are a few key points about the degree of polynomials:

• The degree determines the number of roots a polynomial can have.
• It influences the end behavior of the polynomial's graph.
• The polynomial's degree is actually the sum of the exponents if there are multiple variables.

Identifying the degree of polynomials accurately allows you to classify and work with various polynomial equations effectively.

Standard Form of Polynomial

The standard form of a polynomial is a way of writing it where the terms are ordered by the descending power of the variable. This form facilitates easier reading and comparison of polynomials. In standard form, the term with the highest degree goes first, followed by others in descending order. An example would be writing \(20m^3\) as the leading term since it already satisfies the standard form criteria because it comprises just one term.

• Typically starts with the highest degree term.
• Simplifies addition and subtraction of polynomials.
• Helps reveal the structure of the polynomial.

Knowing how to write and recognize polynomials in standard form assists in various algebraic operations and enhances understanding of polynomial expressions.