Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression, or explain why the expression cannot be simplified. $$30 x^{3}-29 x^{3}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{3}\).

Step 1: Identify Like Terms

In the given expression, the like terms are \(30 x^{3}\) and \(-29 x^{3}\). These terms have the same variable part which are raised to the same power, hence they are like terms.

Step 2: Combine the Coefficients of the Like Terms

Subtract the two coefficients that is \(30 - 29 = 1\).

Step 3: Re-write the Expression

Replace the like terms in the expression with their combined coefficient, which forms the simplified expression: \(1x^{3}\) or simply \(x^{3}\).

Key Concepts

Like Terms

Like terms are critical when working with algebraic expressions. They are terms that have the same variable component raised to the same power. For example, in the expression \(30x^3 - 29x^3\), both terms are like terms because they share the variable \(x\) with the exponent 3.
Recognizing like terms is the first step in simplifying expressions. They make calculations easier because they can be combined through arithmetic operations, such as addition or subtraction. Unlike terms, however, have different variables or powers and cannot be combined in the same straightforward manner.

• Example of like terms: \(5x^2\) and \(-3x^2\)
• Example of unlike terms: \(2x^3\) and \(4xy^2\)

Simplification

Simplification in algebra involves rewriting an expression in a more concise form without changing its value. Simplified expressions look cleaner and are easier to work with for further calculations. The key to simplification is identifying and appropriately combining like terms.
This process often involves basic arithmetic on the coefficients of like terms. For example, in the expression \(30x^3 - 29x^3\), the simplification involves recognizing that the expressions can be unified into a single term by combining their coefficients. Through simplification, the expression becomes more manageable and ready for further application.
Remember:

• Check all components of the expression.
• Focus on like terms when simplifying.
• Always perform arithmetic operations carefully on coefficients.

Combining Like Terms

Combining like terms is a fundamental process in algebra that helps reduce expressions to simpler forms. It applies when dealing with terms that share the same variables and exponents. To combine them, simply add or subtract their coefficients.
In the exercise we are reviewing, combining like terms works like this: the terms \(30x^3\) and \(-29x^3\) are like terms. The coefficients 30 and -29 can be directly combined because their variable parts \(x^3\) are identical. By performing the operation \(30 - 29\), we get a new coefficient of 1, simplifying the expression to \(1x^3\) or simply \(x^3\).

• Always verify that terms are indeed like terms.
• Combine only the coefficients, leaving the variable part unchanged.

Coefficients

Coefficients are the numerical part of a term in an algebraic expression. They dictate the magnitude of the term in conjunction with its variable portion. In the expression \(30x^3\), 30 is the coefficient. Similarly, in \(-29x^3\), -29 is the coefficient.
When simplifying expressions, focus on the coefficients of like terms. Operations involving coefficients impact the overall expression but not the variables themselves. In our example, subtracting the coefficients of \(30x^3\) and \(-29x^3\) resulted in a new coefficient of 1, leading to the simplified term \(x^3\).
Key points:

• Coefficients multiply the variable part of the term.
• They show how many times the base term is counted.
• Operations on coefficients are direct mathematical computations.