Question

Write the coefficient of each term. Assume that letters from the beginning of the alphabet \((a, b, c, \ldots)\) are constants.$$1x$$

Short answer

The coefficient of the term 1x is 1.

Step-by-step solution

Identify the Coefficient

The coefficient is the numerical factor of the term that contains a variable. In the term the variable is 'x' and the numerical factor is '1'.

Key concepts

Algebraic Terms

In algebra, terms are the building blocks of expressions. An algebraic term is a combination of numbers, variables (letters that represent unknown or varying quantities), and sometimes exponents that are multiplied together.

For instance, consider the term '3x^2y'. Here '3' is the coefficient, 'x' and 'y' are variables, and '2' is the exponent applied to the variable 'x'. Together, they form a single algebraic term. When we talk about the structure of such terms, we refer to the coefficient, the variables, and the exponent (if any) as essential components that provide the term with its distinct properties.

Algebraic terms can be combined through addition, subtraction, multiplication, and division to form an algebraic expression. Whenever we see terms separated by a plus or minus sign, it means those terms are part of an algebraic expression. It is important to be able to recognize and work with individual terms to simplify expressions and solve algebraic equations efficiently.

Numerical Factors

A numerical factor, also known as a coefficient, is a number that multiplies a variable within an algebraic term. It tells you how many times to multiply the variable. One key point to remember is that every algebraic term has a coefficient, sometimes explicitly shown and sometimes not.

If we have the term '1x', '1' is the coefficient. You might be wondering why write '1' when it's just 'x'? It is because mathematically, every variable has a coefficient - if one is not shown, it is understood to be '1'. The coefficient '1' might seem redundant, but it's essential when learning the principles of algebra. It reinforces the concept that variables are quantities that can be multiplied, just like numbers.

Moreover, coefficients can be positive or negative, whole numbers, fractions, or decimals. They are crucial in determining the value of the term when the variable amounts are known.

Variables in Algebra

Within algebra, variables are symbols, usually letters, that are used to represent unknown or changeable values. They are the most important part of algebraic terms that differentiate algebra from simple arithmetic.

Variables can stand for numbers of any kind: whole numbers, decimals, fractions, or negative numbers. For instance, 'x' in the term '1x' can be replaced with any numerical value, which makes algebra a powerful tool in representing real-world problems where certain values are unknown or changing.

The beauty of variables in algebra is their versatility; they allow us to create general solutions to problems, as opposed to arithmetic which generally deals with specific numbers and operations. Understanding how to manipulate variables is integral to mastering algebra and unlocking the full potential of mathematical problem-solving.