Problem 29 - Elementary Algebra

Question

For the expressions in the following problems, write the number of terms that appear and then list the terms. $$ 4 y^{3} $$

Step-by-Step Solution

Answer

Expression: \(4y^3\) Answer: There is 1 term in the given algebraic expression, and that term is \(4y^3\).

Step 1: Identify the terms in the expression

The given expression is: $$ 4y^3 $$ In this expression, there is only one term: \(4y^3\).

Step 2: Count the terms

Since there is only one term in the expression, the number of terms is 1.

Step 3: List the terms

As we identified in Step 1, the only term in the given expression is \(4y^3\). Therefore, the list of terms is: $$ 4y^3 $$

Key Concepts

Terms in Algebra

In algebra, a term is a single mathematical expression that can be a number, a variable, or the product or quotient of numbers and variables. Terms are the building blocks of algebraic expressions. Understanding what constitutes a term is crucial as it helps in organizing and simplifying expressions.

• Constants: These are numbers on their own, like 5 or -3.
• Variables: Symbols that stand for unknown values, such as \(x\) or \(y\).
• Coefficients: Numbers preceding variables, like the 4 in \(4y^3\).
• Products of variables and coefficients: Such as \(4y^3\), where 4 is a coefficient and \(y^3\) is the variable part.

When you look at an algebraic expression, separating it into individual terms helps in simplifying and solving it.

Polynomial Expression

A polynomial expression is a sum of terms, where each term is composed of variables raised to whole number exponents and has coefficients. The importance of polynomials lies in their application across various fields of mathematics and science.
Polynomials can vary in complexity based on the number of terms they contain. Here is a simplified breakdown:

• Monomial: An expression with a single term, like \(4y^3\).
• Binomial: An expression with two terms, such as \(3x + 2y\).
• Trinomial: An expression with three terms, like \(x^2 + 3x + 2\).

The degree of a polynomial is determined by the highest power of the variable present. In \(4y^3\), it is 3, making it a third-degree polynomial.

Single-term Expression

A single-term expression, also known as a monomial, features only one term. This simplicity makes them easier to work with compared to complex polynomials. In mathematics, terms like \(7x\) or \(-3z^{4}\) are single-term expressions.Key characteristics include:

• Contains no addition or subtraction between terms.
• Is composed of just one piece with variables, constants, or both.
• The degree of a monomial is the sum of the exponents of the variables; in \(4y^3\), the degree is 3.

Understanding single-term expressions sets the foundation for grasping more complex algebraic structures.