Problem 31 - Beginning and Intermediate Algebra

Question

In each term, give the numerical coefficient. \(-0.5 x^{3}\)

Step-by-Step Solution

Answer

-0.5

Step 1: Identify the term

The term given is \(-0.5 x^{3}\).

Step 2: Locate the numerical coefficient

The numerical coefficient is the number that is multiplying the variable \(x^{3}\). Look closely at the term.

Step 3: Extract the numerical coefficient

In the term \(-0.5 x^{3}\), the numerical coefficient is \-0.5\.

Key Concepts

term identification

Understanding terms in algebraic expressions is essential for solving them correctly. In an algebraic expression, a term is a single mathematical phrase, which can be a number, a variable, or the product of numbers and variables.
For example, in the term \(-0.5 x^{3}\), we have a single expression that combines a numerical factor and a variable raised to a power.
Identifying terms correctly is the first step in understanding more complex algebraic expressions. Remember that terms are separated by plus or minus signs. The presence of a term indicates that we are dealing with a mathematical entity that forms part of an equation or expression.

algebraic expressions

Algebraic expressions consist of numbers, variables, and operators (like + and -) combined together in a meaningful way.
They can vary from simple expressions such as \(-0.5 x^{3}\) to very complex ones incorporating multiple terms and operations. In our example, \(-0.5 x^{3}\) is an expression where \-0.5\ is the coefficient and \x^{3}\ is the variable part.
Understanding algebraic expressions is crucial for solving equations, simplifying terms, and performing operations. The key elements in any algebraic expression are:

• Numerical coefficients
• Variables
• Exponents

In higher-level algebra, these elements combine in more sophisticated ways, but the fundamental principles remain the same.

multiplying variables

Multiplying variables involves combining similar terms by using multiplication. This means you multiply the coefficients and add the exponents of the same variables.
In our example \(-0.5 x^{3}\), the variable x is raised to the power of 3, meaning that x is being multiplied by itself three times. When multiplying this term by another term with an x variable, you add the exponents.
For example, \(-0.5 x^{3} * 2x^{2} = -1 x^{5}\). Here is how it works:

• Multiply the coefficients: \-0.5 * 2 = -1\
• Add the exponents: \3 + 2 = 5\

So the result is \-1 x^{5}\. By practicing these steps and understanding the underlying principles, multiplying variables becomes easier and more intuitive.