Problem 22 - Introductory Algebra for College Students

Question

In Exercises \(15-28,\) simplify each algebraic expression, or explain why the expression cannot be simplified. $$29 x^{2}-30 x^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Identifying Like Terms

The expression has like terms \(29x^{2}\) and \(-30x^{2}\). Like terms are algebraic expressions having the same variable and exponent.

Step 2: Simplifying Like Terms

Combine the like terms by subtracting the coefficients of the like terms. That means \(29 - 30\) which results to \(-1\). Since there is no other term left in the equation, the answer is \(-1x^{2}\). However, as a general rule in mathematics, when the coefficient is -1, it is usually not written out. Hence, the answer simplifies to \(-x^{2}\).

Key Concepts

Like Terms

Understanding like terms is foundational in simplifying algebraic expressions. Like terms are terms within an expression that have exactly the same variable parts—that is, they have the same variables raised to the same powers. For instance, in the terms \(2x^3\) and \(-5x^3\), both have the variable part \(x^3\). This similarity allows them to be combined through addition or subtraction.

However, terms like \(2x^3\) and \(2x^2\) are not like terms because their exponents differ; one is cubed while the other is squared. In simpler terms, think of like terms as family members that share the same surname. This commonality makes it easy to group them together, which is a key step in the process of simplification.

Combining Like Terms

Once you've identified the like terms, the next step is combining like terms. Combining involves performing addition or subtraction on the coefficients (the numerical parts) of the like terms while keeping the variable part intact.

Consider the terms \(7x\) and \(-3x\). Since they are like terms (both have the variable \(x\)), we focus only on the numbers in front: \(7\) and \(-3\). Adding these together, \(7 + (-3)\), gives us \(4x\). That’s the essence of combining like terms: simplifying the expression to make it more manageable and more understandable. It's a bit like consolidating debts; you’re bringing together what you owe to have a clearer view of the total.

Algebraic Expression Simplification

Finally, the process of algebraic expression simplification can be viewed as cleaning up an equation or expression to its simplest form without changing its value. This involves identifying and combining like terms, applying the distributive property where necessary, and performing any operations required by the expression.

Back to our textbook example \(29x^{2} - 30x^{2}\), after identifying \(29x^{2}\) and \(-30x^{2}\) as like terms, you subtract the coefficients (29 and 30) to get \(-1\), which leaves us with \(-1x^{2}\) or more simply, \(-x^{2}\). Simplification makes expressions less complicated, easier to work with in equations, and often helps in revealing the more practical or useful form of the mathematical statement.