Problem 7 - Elementary Algebra Within Reach

Question

In Exercises 5-10, identify the terms of the expression. $$ \frac{5}{3}-3 y^{3} $$

Step-by-Step Solution

Answer

The terms of the expression \(\frac{5}{3} - 3y^3\) are \(\frac{5}{3}\) and \(-3y^3\).

Step 1: Identify the First Term

In the expression \(\frac{5}{3} - 3y^3\), the first term is \(\frac{5}{3}\). This term consists only of a number.

Step 2: Identify the Second Term

The second term in the expression is \(-3y^3\). We include the negative sign with this term because terms are separated by plus or minus signs. Hence, if a minus sign precedes a term, we consider it as part of that term.

Key Concepts

Terms of an Expression

An expression in algebra is like a phrase made up of different parts, and these parts are called "terms." Terms are separated from each other by plus or minus signs and can be numbers, variables, or a combination of both.
In the expression \(\frac{5}{3} - 3y^3\), there are two terms:

• \(\frac{5}{3}\)
• \(-3y^3\)

These terms come together to form the entire expression. Understanding that each part operates separately is key to simplifying or evaluating expressions.

Polynomial Terms

A polynomial is a type of expression with special terms known as polynomial terms. Polynomial terms are formed from constants and variables, usually by way of multiplication. Each term in a polynomial can be classified based on its degree, which is determined by the sum of the exponents of the variables.
In the expression \(-3y^3\), the term is considered a polynomial term:

• The variable part consists of \(y^3\), which makes it a term of degree 3.
• The numeral part, \(-3\), acts as the coefficient of this term.

Terms in polynomials can vary widely, but they are always the building blocks of polynomial expressions.

Constant Term

A constant term in an expression is a term that does not contain any variables. It remains unchanged no matter what value the variable might take.
In the expression \(\frac{5}{3} - 3y^3\), the constant term is \(\frac{5}{3}\):

• This term is a number on its own, completely independent of any variable.
• Constant terms represent fixed values in every trial or calculation involving the expression.

Unlike variable terms or polynomial terms, the constant term doesn’t change as the variables alter.

Variable Term

Variable terms are parts of an expression that contain variables, which are symbols that represent numbers. These terms can change as the variables change values. In the expression \(\frac{5}{3} - 3y^3\), \(-3y^3\) is the variable term:

• Here, the variable is \(y\), and it is raised to the power of 3, so it depends on the value of \(y\).
• The coefficient, \(-3\), multiplies the entire variable portion.

Variable terms play a crucial role in equations and allow expressions to describe a range of possible scenarios.