Problem 3 - Intermediate Algebra

Question

Fill in the blanks. A term, such as \(9,\) that consists of a single number is called a ______ term.

Step-by-Step Solution

Answer

A constant term.

Step 1: Identify the Type of Term

A term in mathematics can be either a single number, a variable, or a combination of both (a coefficient and a variable). In this exercise, the term is just a single number: 9.

Step 2: Recognize the Definition

When a term consists of just a single number without any variables, it is given a specific name in mathematics. This name indicates its simplicity as it does not contain a variable part.

Step 3: Fill in the Blank

A single number term is referred to as a 'constant' because it does not change; it remains the same value. Therefore, the correct term to fill in the blank is 'constant.'

Key Concepts

Single Number

In mathematics, a single number by itself is known as a 'constant term.' This is because its value does not change. Unlike terms that contain variables, which can vary based on different conditions, single numbers stay the same. For example, the number 9 is always 9, no matter what. These constant terms are the building blocks of more complex mathematical expressions.

• A constant like 9 is easy to work with because it's just that number.
• You don't have to change or manipulate it based on other factors.
• It provides a stable reference point when performing calculations.

Understanding constant terms helps students ground themselves in math, providing a familiar starting point amid more complicated expressions.

Mathematical Term

A mathematical term can be thought of as a single piece of a mathematical expression. Terms can be numbers, variables, or a combination of both. Each term is separated by plus or minus signs in an expression. For instance, in the expression \(3x + 5y - 2\), there are three distinct terms: \(3x\), \(5y\), and \(-2\).

• Terms are often combined to form algebraic expressions.
• They can represent simple values or complex combinations of numbers and variables.
• Recognizing different types of terms helps in organizing and simplifying expressions.

Understanding what constitutes a term is crucial in manipulating and solving equations effectively.

Variable Coefficient

Variable coefficients play a key role in algebraic expressions. A coefficient is the number that directly multiplies a variable in a term. Think of it as the weight or size of the variable it multiplies. For example, in the term \(5x\), 5 is the coefficient of the variable \(x\). This tells you that for every 1 unit of \(x\), the term contributes 5 to the total.

• Coefficients can be positive or negative, impacting the direction of a graph in a function.
• They are essential in scaling the influence of variables within an expression.
• Identifying coefficients helps in operations like factoring and simplifying expressions.

Grasping the idea of a variable coefficient is essential for deeper understanding in algebra and beyond.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. These expressions can range from simple to quite complex. At their core, they are used to express calculations compactly. In an equation like \(2x + 3 = 11\), the left side depicts an algebraic expression.

• Algebraic expressions provide a way to model real-world scenarios mathematically.
• Different terms within the expression are manipulated to find solutions to equations.
• They can be expanded, factored, simplified, or solved depending on the problem at hand.

Understanding and working with algebraic expressions is foundational to algebra, allowing students to solve problems and predict outcomes based on mathematical models.