Question

Write the coefficient of each term. Assume that letters from the beginning of the alphabet \((a, b, c, \ldots)\) are constants.$$3 c x^{3}$$

Short answer

The coefficient of the term \(3 c x^{3}\) is \(3c\).

Step-by-step solution

Identify the Terms

In the expression \(3 c x^{3}\), there is only one term, which is the entire expression.

Determine the Coefficients

The coefficient of the term is the numerical factor that is multiplied by the variables. In this case, the term \(3 c x^{3}\) has \(3c\) as its coefficient, since \(3\) is the number and \(c\) represents a constant, both are multiplied by the variable \(x^{3}\).

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations such as addition, subtraction, multiplication, and division. Think of them as phrases that convey mathematical ideas. These expressions can be simple, with just one term, like 5x or more complex with several terms, like 5x - 3y + 2z. Each part of an algebraic expression that's added or subtracted is called a term. A term can be a single number (which is called a constant because its value doesn't change), a single variable (x, y, z), or numbers and variables multiplied together.

When dealing with algebraic expressions, it's critical to understand what each part represents in order to work with and simplify these expressions effectively. This also forms the basis for solving algebraic equations, where we find the value of the variables that make the equation true.

Coefficients

Coefficients are like the 'multiplier' of the variables within an algebraic expression. They provide a value by which the variable is scaled. If you see 2x, the '2' is the coefficient, indicating that x is multiplied by 2. In the term 3 c x^{3}, our coefficient is 3c. This might look a bit different because here, c is not a variable; it's a constant. Remember, letters at the beginning of the alphabet often represent known constants, while letters towards the end (like x, y, z) typically signify variables. So, we're multiplying the variable x^{3} (x cubed) by the constant c and then by the number 3. Identifying coefficients correctly is crucial for simplifying expressions and solving algebraic equations.

Constants and Variables

Constants and variables are the nuts and bolts of algebra. Constants are fixed values. They don't change as the expression is manipulated. In the expression 3 + a, the number '3' is a constant; its value is always three. On the flip side, variables are the letters or symbols that represent unknown values. In 3 + a, 'a' is a variable; its value can vary depending on the context.

When we look at a term like 3 c x^{3}, it's vital to distinguish between the constants and variables. Here, c is considered a constant and x is the variable. Being clear on what constitutes a constant and what is a variable helps in performing algebraic operations correctly, such as addition and multiplication, and it's essential when you start solving equations. For instance, when you're combining like terms, you'd only combine terms with the same variable part—that's where knowing your constants and variables becomes super important.