Problem 28 - Introductory Algebra for College Students

Question

In Exercises \(15-28,\) simplify each algebraic expression, or explain why the expression cannot be simplified. $$34 x^{2}-x^{2}$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \(34x^{2} - x^{2}\) is \(33x^{2}\).

Step 1: Identify like terms

In the algebraic expression \(34x^{2} - x^{2}\), the like terms are \(34x^{2}\) and \(-x^{2}\). They are considered like terms because they have the same variable \(x\) raised to the same power \(2\).

Step 2: Combine like terms

In algebra, when we have like terms, we add or subtract the numerical coefficients (the numbers in front of the variables). Here, the algebraic expression \(34x^{2} - x^{2}\) becomes \(33x^{2}\), because \(34 - 1 = 33\).

Step 3: State the simplified expression

So, the algebraic expression \(34x^{2} - x^{2}\) simplifies to \(33x^{2}\).

Key Concepts

Like Terms

In algebra, understanding the concept of like terms is fundamental. Like terms are terms that contain the same variables raised to the same exponent.
This means that the only differences between them can be the coefficients or the numbers in front of these variables.
For example, in the expression \(34x^2 - x^2\), both terms have the variable \(x\) raised to the power of 2, making them like terms.

Like terms allow us to simplify expressions by performing operations on the coefficients.
When we recognize like terms, it means we can start simplifying an expression by adding or subtracting their coefficients, streamlining the expression considerably.
Focusing on identifying like terms is the starting point for simplifying algebraic expressions effectively.

Combining Coefficients

Once we've identified the like terms in an algebraic expression, the next step is to combine their coefficients.
The coefficient is simply the numerical part that multiplies the variable part of a term. For instance, in the term \(34x^2\), the number 34 is the coefficient.

To combine coefficients, you need to add or subtract these numbers, depending on the operation involved between the terms.

• In the expression \(34x^2 - x^2\), the coefficients are 34 and -1, respectively.
• Since subtraction is indicated, you subtract: \(34 - 1 = 33\).

By combining the coefficients, the expression \(34x^2 - x^2\) simplifies to \(33x^2\).
Combining coefficients effectively reduces the complexity of the expression, making it easier to handle or further evaluate.

Algebra for College Students

College students often encounter algebraic expressions that appear complex at first glance.
However, breaking down these expressions into simpler forms is a skill honed by understanding foundational concepts like identifying like terms and combining coefficients.

Algebra at the college level involves not just solving equations but also applying algebraic reasoning to various fields of study.

• Simplifying expressions is crucial for tackling equations, inequalities, and even functions.
• Recognizing patterns in expressions is a part of developing problem-solving skills, as it allows you to foresee solutions and make calculations more manageable.

The ability to simplify expressions efficiently enables students to approach problems analytically, paving the way for success in disciplines that rely heavily on algebraic thinking.