Question

Write the coefficient of each term. Assume that letters from the beginning of the alphabet \((a, b, c, \ldots)\) are constants.$$-x$$

Short answer

The coefficient of the term \(-x\) is \(-1\).

Step-by-step solution

Understanding the Terms

A coefficient is a numerical or constant quantity placed before and multiplying the variable in an algebraic term. In this case, the term provided is \-x, which consists of a variable \(x\) and its coefficient.

Identifying the Coefficient

Since the term is \-x and there is no number written before \(x\), it is understood that the coefficient is implied to be \(-1\). The negative sign indicates that the coefficient is \(-1\) times the variable.

Key concepts

Coefficients in Algebra

In algebra, coefficients play a crucial role in defining the relationships between variables within an expression or equation. A coefficient is typically a number, but it can also be a constant term that multiplies a variable. For example, in the algebraic term \(3x\), the number \(3\) is the coefficient of \(x\). It tells us how many times the variable \(x\) is to be counted.

Understanding coefficients is essential as it impacts how an algebraic expression can be manipulated and simplified. It's common for students to overlook the coefficient when it is \(1\) or \(\-1\), as these are often implied and not explicitly written. For example, the term \(\-x\) from our exercise implies a coefficient of \(\-1\) even though it's not visibly present next to the variable \(x\).

Constant and Variable Terms

Algebraic expressions consist of constant and variable terms which represent numbers and quantities. Variable terms include letters that stand for unspecified numbers and can change, whereas constant terms are fixed numbers that do not vary.

For instance, in \(2x + 5\), \(2x\) is a variable term, and \(5\) is a constant term. The variable term includes a coefficient (2 in this case) and a variable (x), while the constant term is simply a number standing alone. Recognizing the difference between these types of terms is vital for solving algebraic problems as it helps in performing operations like combining like terms and simplifying expressions.

Interpreting Algebraic Expressions

Interpreting algebraic expressions involves understanding the components and the structure of the expression to determine its value or simplify it. Each expression is made up of terms that can be broken down into coefficients, variables, and constants.

Consider the algebraic expression \(3a + 4b - c\). To interpret this expression correctly, you need to recognize each term and its coefficient. The term \(3a\) has a coefficient of \(3\), \(4b\) has a coefficient of \(4\), and \(\-c\) implies a coefficient of \(\-1\) because of the negative sign and the absence of any other numbers. Interpreting expressions allows a student to approach algebraic operations with precision, such as adding or subtracting like terms and factoring.