Problem 14 - Elementary Algebra

Question

Determine the coefficients. In the term \(6 x^{3}\), the coefficient of (a) \(x^{3}\) is _________. (b) 6 is _________.

Step-by-Step Solution

Answer

Answer: The coefficient of x^3 is 6, and the coefficient of 6 is 1.

Step 1: Identify the given term

The given term is 6x^3. Here, 6 is a constant and x^3 is a variable raised to the power of 3.

Step 2: Determine the coefficient of x^3

In the term 6x^3, the coefficient of x^3 is the constant that is multiplied by x^3. Here, that constant is 6. Therefore, the coefficient of x^3 is 6.

Step 3: Determine the coefficient of 6

In the term 6x^3, when we refer to the coefficient of 6, we are asking for the constant factor that is multiplied by 6 to give the term. 6 is already a constant, and there is no other numerical factor to multiply it by. Therefore, the coefficient of 6 is 1. So, the coefficients are: (a) The coefficient of x^3 is 6. (b) The coefficient of 6 is 1.

Key Concepts

Algebraic Terms

Algebraic terms are the building blocks of algebraic expressions. Each term is made up of different parts, typically constants and variables. A single term can include:

• A constant, which is a fixed numerical value, like 6 or 2.
• A variable, represented by a letter such as \(x\), \(y\), or \(z\), which can take different values.
• A coefficient, which is a number multiplied by the variable, indicating how many times the variable is to be considered.

In the term \(6x^3\), we see that "6" is the coefficient, and \(x^3\) is the variable component. This term combines all three elements effectively. The parts of an algebraic term work together to contribute to forming an algebraic expression, like \(3x+4y-7\). By understanding each piece clearly, any algebraic expression can be broken down and analyzed effectively.

Constant

A constant is a value that does not change. It remains the same regardless of the context or the values of variables within an expression. In algebraic terms, a constant stands alone without any variables attached to it. It acts as a fixed quantity.
For example, consider the term \(6x^3\). In this term, the number 6 acts as a constant. It is important to note that constants are not influenced by operations conducted on variables in an algebraic expression.
In equations, constants provide a reference or baseline value. This helps in solving for any unknown variables. Constants are crucial as they anchor the expressions, giving them a numerical basis from which other parts can fluctuate or vary.

Variable

Variables are the letters or symbols used in mathematical expressions to represent unknown or changeable numbers. They allow us to form general mathematical statements and equations. Variables serve as placeholders that can take on varied numerical values.

• Common variables used include letters like \(x\), \(y\), and \(z\).
• They can be raised to a power, such as \(x^3\), which indicates that the variable is multiplied by itself a certain number of times.
• Variables make it possible to describe and solve problems involving quantities that are not fixed.

In the expression \(6x^3\), \(x\) is the variable, and it is raised to the third power, indicating it is multiplied by itself twice. Variables are indispensable in algebra as they enable mathematicians and students to write formulas and equations that apply to numerous situations.