Question

Write the coefficient of each term. Assume that letters from the beginning of the alphabet \((a, b, c, \ldots)\) are constants.$$2 a x^{5}$$

Short answer

The coefficient of the term is 2a.

Step-by-step solution

Identify the Coefficients

The coefficient of a term in an algebraic expression is the numerical factor before the variable(s). It includes numbers and any constant variables (like a, b, etc.). In the given term, we need to identify the coefficient attached to the variable part, which in this case is a power of x.

Key concepts

Coefficient Identification

Understanding coefficients in algebra is essential for working with algebraic expressions. A coefficient is a number or constant that multiplies a variable. For instance, in the term \(2ax^5\), the coefficient is the product of the constant \(2\) and the constant variable \(a\). To identify the coefficient in any algebraic term, look for the factors that are not part of the exponent.

In the provided step by step solution, identifying the coefficient requires recognizing that the \(x\) raised to the fifth power is the variable component and the product \(2a\) before it represents the coefficient. Even though constants like \(a\), \(b\), \(c\), etc., may not have specific numerical values in the expression, they still form part of the coefficient if they precede a variable.

By pinpointing the parts of the algebraic term, we can understand and manipulate expressions to solve for unknown variables, simplify expressions, or perform operations such as factoring. This fundamental skill lays the groundwork for all further studies in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and constants, connected by mathematical operators such as addition, subtraction, multiplication, and division. To comprehend an algebraic expression, such as \(2ax^5\), it's important to identify its components. The variables (often represented by letters), such as \(x\), stand in for unknown quantities, and the exponents (like \(5\) in \(x^5\)) indicate how many times the variable is multiplied by itself.

Grasping the structure of algebraic expressions allows students to perform a variety of operations with them. It's essential to recognize that each term in an expression can include numbers, variables, or both, and that the coefficient is intimately tied to its corresponding variable. Algebraic expressions are the building blocks for equations and formulate the language in which many real-world problems can be mathematically modeled and solved.

Exponents in Algebra

Exponents play a critical role in algebra, indicating repeated multiplication of a base number or variable. The number \(5\) in \(x^5\) is an exponent, and it means that \(x\) is used as a factor five times: \(x*x*x*x*x\). When dealing with algebraic expressions that include exponents, it's vital to remember the laws of exponents as they help simplify expressions, solve equations, and understand the expression's behavior.

Exponents can apply to both numerical and variable components of terms in algebraic expressions. The power to which a variable is raised can significantly affect the solution to an equation or the simplification process. Working comfortably with exponents includes mastering operations such as multiplying powers with the same base, raising a power to a power, as well as understanding zero and negative exponents. Acknowledging these properties ensures that algebraic manipulation becomes more intuitive and efficient.