Problem 18 - Introductory Algebra for College Students

Question

In Exercises \(15-28,\) simplify each algebraic expression, or explain why the expression cannot be simplified. $$14 x^{3}+8 x^{3}$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \(14 x^{3}+8 x^{3}\) is \(22x^{3}\)

Step 1: Identify Similar Terms

In the algebraic expression \(14 x^{3}+8 x^{3}\), the similar terms are \(14x^{3}\) and \(8x^{3}\). Both terms have the same variable part, \(x^{3}\), and their coefficients are 14 and 8 respectively.

Step 2: Use Distributive Property

The distributive property of multiplication over addition allows us to combine similar terms. As per distributive law, we get \(14x^{3} + 8x^{3} = (14 + 8)x^{3}\)

Step 3: Calculate the Coefficient

The value after adding coefficients of the similar terms is obtained by adding 14 and 8 which equals 22. Thus, the simplified form of the algebraic expression is \(22x^{3}\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions. It allows us to multiply a single term by two or more terms inside a parenthesis. This property states that for any numbers a, b, and c:

• \( a(b + c) = ab + ac \)

When simplifying expressions, the distributive property helps to combine like terms by applying multiplication and addition in a more straightforward manner. For example, if we have the expression \( (14 + 8)x^3 \), using the distributive property, we distribute the multiplication of the term outside the parenthesis to each term inside, giving us \( 14x^3 + 8x^3 \). This way, we can more easily add the coefficients.
This property is especially useful when dealing with polynomials or any expressions that require reorganizing terms for simplicity.

Like Terms

Like terms in algebra are terms that have the same variable part. This means they have identical variables raised to the same exponents. In other words, they "look" alike except for their coefficients. Consider the following characteristics of like terms:

• The terms \(14x^3\) and \(8x^3\) are like terms because they share the same variable \(x\) raised to the same power, which is \(3\).
• In the expression \(14x^3 + 8x^3\), both terms can be combined because their variable parts are identical.
• They make it possible to perform addition or subtraction directly on the coefficients.

Understanding like terms is crucial for simplifying expressions because it determines which terms can be combined. Without recognizing like terms, simplification would be impossible, as the terms would mistakenly appear unrelated.

Coefficients in Algebra

Coefficients are numerical or constant factors placed in front of variables in algebraic expressions. They quantify the variable part of a term, acting as multipliers. Here's what you need to know about coefficients:

• In the expression \(14x^3\), the number 14 is the coefficient.
• The function of coefficients is to indicate how many times the variable term is being added together. For instance, \(14x^3\) means that \(x^3\) is being multiplied by 14.
• When combining like terms, you simply add or subtract their coefficients while keeping the variable part unchanged. In our example, adding the coefficients 14 and 8 gives us 22, resulting in the term \(22x^3\).

Recognizing coefficients allows us to handle algebraic operations more effectively, ensuring the expressions are simplified without altering their fundamental meanings.