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Chapter 13: Problem 21

Find the degree of each polynomial. $$12 c^{3}$$

Short Answer

Expert verified
The degree of the polynomial \(12c^3\) is 3.

Step by step solution

01

Identify the Polynomial

The given polynomial is written as \(12c^3\). It consists of one term, which is known as a monomial.
02

Determine the Exponent in the Term

For each term in the polynomial, identify the exponent of the variable. In this case, the variable is \(c\) and the exponent is \(3\).
03

Identify the Degree of the Polynomial

The degree of a polynomial is determined by the highest exponent of the variable in the polynomial. Since \(12c^3\) is a monomial, the degree is the exponent \(3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of Polynomial
The degree of a polynomial is a crucial concept in understanding algebraic expressions. It helps us describe the power to which the variable is raised. In simpler terms, the degree tells us the highest exponent present in the polynomial when expressed in its standard form. For the polynomial given as \(12c^3\), the degree is determined by looking at the exponent of the variable \(c\), which is \(3\). Therefore, the degree of this polynomial is \(3\). This indicates that the polynomial will behave with a cubic-like growth pattern.
Exponents
In algebra, exponents are used to express powers of a number or variable. Understanding exponents is vital as they inform us how many times a number is multiplied by itself. For instance, in \(c^3\), the exponent is \(3\), suggesting that \(c\) is multiplied by itself three times: \(c \times c \times c\).

Exponents are a form of shorthand notation that simplifies complex multiplication. When dealing with polynomials, identifying the exponent allows you to determine the degree easily, which is essential in solving and analyzing algebraic problems.
Monomial
A monomial is a polynomial with only one term. It is one of the simplest types of polynomials and usually involves a constant, variable, or both, multiplied together. For example, \(12c^3\) is a monomial with a constant coefficient of \(12\) and a variable \(c\) raised to an exponent \(3\).

In monomials, the degree is straightforward to determine because it will always be the exponent of the variable present in the term. This makes monomials an excellent starting point when learning about polynomials and their degrees.
Algebra Concepts
Algebra is the study of mathematical symbols and the rules for manipulating these symbols. At the core of algebra are different concepts that form the building blocks for solving equations and understanding relationships within mathematical problems.

Some basic algebra concepts include:
  • Variables: Symbols that represent numbers or values, for example, \(c\). They are essential in forming equations and functions.
  • Constants: Fixed values that do not change, like the number \(12\) in our given polynomial.
  • Operations: Include addition, subtraction, multiplication, and division.
  • Expressions: Combinations of variables, constants, and operations, for example, \(12c^3\).
Grasping these elementary concepts can greatly enhance one's ability to tackle more complex algebraic challenges.

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