Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression, or explain why the expression cannot be simplified. $$26 x^{2}-27 x^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(-1x^{2}\).

Step 1: Identify Like Terms

In an algebraic expression, 'like terms' are the terms with the same variable raised to the same exponent. In our case, \(26 x^{2}\) and \(-27 x^{2}\) are like terms because they both have the variable \(x\) raised to the power of \(2\).

Step 2: Combine Like Terms

Like terms can be combined by addition or subtraction. Here, the term \(27 x^{2}\) is subtracted from \(26 x^{2}\), yielding \(-1x^{2}\). Note here, we have calculated \(26 - 27 = -1\). So, the expression simplifies to \(-1x^{2}\).

Key Concepts

Like Terms

In the world of algebra, understanding like terms is crucial for simplifying expressions. Like terms are terms within an algebraic expression that have the same variable, or variables, raised to the same power.
For example, in the expression given, both terms have the variable \(x\) raised to the power of \(2\). As a result, \(26x^2\) and \(-27x^2\) qualify as like terms.
Recognizing these is the first step in simplifying any algebraic expression. Identifying like terms correctly is essential because it allows us to

• Combine terms efficiently
• Avoid mistakes in simplification

This fundamental concept applies to any expression and sets the stage for further operations.

Simplification

Simplification is the process of reducing an algebraic expression to its simplest form. The goal is to make the expression easier to understand and work with.
For any given expression, this typically involves identifying and combining like terms or reducing the coefficients.
In our case, the given expression, \(26x^2 - 27x^2\), simplifies to \(-1x^2\).

• The terms \(26x^2\) and \(-27x^2\) are combined because they are like terms.
• Perform the operation of subtraction \(26 - 27 = -1\).

Once simplified, the expression is in its simplest arithmetic form, making it easier to understand and use in further calculations or applications.

Combining Terms

Combining like terms is a key skill in algebra that helps streamline expressions into simpler forms. This operation involves either adding or subtracting the coefficient of like terms while retaining the variable and its exponent.
For our example, you subtract the coefficients of each respective like term:

• The coefficient of \(26x^2\) is \(26\).
• The coefficient of \(-27x^2\) is \(-27\).

By subtracting \(27\) from \(26\), we get \(-1\), resulting in the expression \(-1x^2\).
Combining terms can significantlyease further computations and help understand the core structure and value of an expression.