Problem 16 - Introductory and Intermediate Algebra for College Students 4th

Question

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 expression is \(24 x^{2}\).

Step 1: Identify like terms

In the given expression, \(6 x^{2}\) and \(18 x^{2}\) are like terms. This is because they contain the same variable, \(x\), raised to the same power, 2.

Step 2: Combine like terms

Like terms can be combined by adding or subtracting the coefficients. In this case, we are adding the coefficients as the two terms are connected by a plus sign. So, we add the coefficients 6 and 18 to get 24.

Step 3: Write down the simplified expression

With the combined coefficient of 24, we rewrite the simplified expression as \(24 x^{2}\).

Key Concepts

Like Terms in Algebra

In algebra, 'like terms' are terms that have the same variables raised to the same powers, but they may have different coefficients. For instance, consider the terms 2a and 5a. Both terms have the variable a to the first power, which makes them like terms. However, the terms 2a and 5a^2 are not like terms since the variable a is raised to different powers. It's important to identify like terms when simplifying algebraic expressions, because only like terms can be combined together.

Let's take a look at an example. In the expression 3x^3 + 7x^2 + 4x^3, the terms 3x^3 and 4x^3 are like terms because they both have the variable x raised to the third power. However, the term 7x^2 is not a like term to the others because the exponent on x is different.

Combining Like Terms

Once you have identified the like terms in an algebraic expression, you can simplify it by 'combining' those terms. This process involves adding or subtracting the coefficients of the like terms. Remember, the variable part must not change; only the coefficients are combined. For example, if you are given the expression 4y + 7y, you combine the coefficients (4 and 7) to get 11, resulting in the simplified expression 11y.

In our original exercise, the expression 6x^2 + 18x^2 is simplified by combining the coefficients 6 and 18. Since these are like terms, you add the coefficients to get 24, resulting in a new expression, 24x^2. The process of combining like terms is crucial to simplifying algebraic expressions and solving algebraic equations.

Algebraic Coefficients

In algebra, coefficients are the numerical part of the terms that are multiplied by the variables. For example, in the term 8z, 8 is the coefficient, and z is the variable. Coefficients can be whole numbers, fractions, decimals, or integers, and they provide information about how much of a certain variable is present in a term. Understanding the role of coefficients is key to simplifying algebraic expressions, as they are what you manipulate when you are combining like terms.

Let's consider the coefficient in the exercise we are looking at: the terms 6x^2 and 18x^2 both have x^2 as the variable part, but they have different coefficients, 6 and 18 respectively. When you combine these like terms by adding their coefficients, you're able to simplify the expression to 24x^2, where 24 is the combined coefficient. Coefficients play a fundamental part in shaping the value of algebraic expressions.