Problem 16 - Introductory Algebra for College Students

Question

In Exercises \(15-28,\) simplify each algebraic expression, or explain why the expression cannot be simplified. $$6 x^{2}+18 x^{2}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(6 x^{2}+18 x^{2}\) is \(24 x^{2}\).

Step 1: Identification of Like Terms

Breaking down the algebraic expression \(6 x^{2}+18 x^{2}\), it can be noticed that both terms are like terms. Like terms are terms that have the same variables and powers.

Step 2: Simplifying Exercise with Like Terms

Like terms can be combined together using the basic principles of addition. Adding together \(6 x^{2}\) and \(18 x^{2}\) would then result to \(24 x^{2}\).

Key Concepts

Like Terms in Algebra

When we talk about like terms in algebra, we are referring to terms that have exactly the same variable factors raised to the same powers. For example, in the expression provided from the textbook exercise, both terms, 6x^2 and 18x^2, are considered like terms because the variable x is squared in both instances.

Why is identifying like terms important? It's because like terms can be combined through addition or subtraction to simplify algebraic expressions. Simplification helps in making mathematical problems more manageable and can also reveal the underlying structure of algebraic expressions. When combining like terms, we must ensure to only adjust the coefficients – the numerical part of the terms – while keeping the variable portion unchanged.

Here's a simple guideline for identifying like terms: check if the variables and their exponents match. If they do, you've got like terms that you can simplify to a single term with ease.

Basic Algebraic Operations

Basic algebraic operations involve addition, subtraction, multiplication, and division applied to algebraic expressions. In our exercise example, the operation at hand is addition. However, unlike addition with simple numbers, when we add algebraic expressions, we have to consider the types of terms we are adding together.

As demonstrated in the textbook solution, only like terms can be directly added together in algebra. Their coefficients are added while the variables are not altered. This is akin to saying that if you have 6 apples and someone gives you 18 more apples, you end up with 24 apples altogether. Similarly, 6x^2 plus 18x^2 gives us 24x^2 because the 'apples', in this case, are the x^2 terms. This addition is a fundamental concept and sets the basis for more complex algebraic manipulations.

Adding Polynomials

The process of adding polynomials is an extension of the addition of like terms. A polynomial is a mathematical expression consisting of variables (also called indeterminates), coefficients, and non-negative integer exponents of those variables. It is composed of multiple terms, like the expression in our problem, which is a binomial since it has two terms.

When adding polynomials, each term in one polynomial should be added to the corresponding like term in the other polynomial. If there are no like terms to match a particular term, it simply carries into the solution as is. In our problem, both terms are part of a single polynomial and are like terms, so the operation is straightforward. But remember, if we had a more complex polynomial with different terms, such as both x^2 and x, we would only be able to combine the terms that are alike. The result of adding polynomials is a simpler polynomial without changing the original equation's value.