Problem 1 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$3 x^{3}+5 x^{3}$$

Step-by-Step Solution

Answer

The combined expression is \(8x^3\).

Step 1: Identify Like Terms

The expression given is \(3x^3 + 5x^3\). Notice that both terms include the variable \(x^3\), which means they are like terms.

Step 2: Combine Like Terms

To combine like terms, simply add or subtract the coefficients (the numbers in front of the variables) while keeping the variable part the same. In this case, add the coefficients, which are 3 and 5: \(3 + 5 = 8\). So, \(3x^3 + 5x^3 = 8x^3\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that involve variables raised to whole number exponents, coefficients, and constants. They can appear as a sum of several terms, each term being a product of a coefficient and a variable(s) raised to a power. For example, the expression \(3x^3 + 5x^3\) is a polynomial expression. Here, the terms \(3x^3\) and \(5x^3\) are made up of coefficients (3 and 5, respectively) and a variable part \(x^3\).
Polynomial expressions are ubiquitous in algebra and mathematics as a whole. They serve as a fundamental building block for more complex equations and functions. Recognizing a polynomial expression involves identifying the structure:

• Each component, also known as a term, consists of a coefficient and a variable part.
• The variable part is usually in the form of \(x^n\), where \(n\) is a non-negative integer.
• The expression can have one or more terms.

Understanding polynomial expressions helps in tasks such as finding roots, graphing functions, and solving algebraic equations.

Like Terms

Like terms in algebra are terms that have the same variables raised to the same powers. They can be easily combined because they describe the same kinds of quantities. In the expression \(3x^3 + 5x^3\), both terms are like terms because they contain the same variable, \(x^3\). This means the variable and its exponent match in each term.
Identifying like terms is crucial when simplifying expressions. It involves the following steps:

• Look for terms that have identical variable parts, regardless of their coefficients.
• Ensure the exponent of the variable is the same across these terms.
• Ignore coefficients when identifying like terms; they need to match only the variable part.

By combining like terms, you can more easily manipulate and simplify algebraic expressions, leading to cleaner, more comprehensible solutions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions often involves combining like terms to reduce the expression to its simplest form. The process makes the expression easier to understand and solve. When simplifying expressions like \(3x^3 + 5x^3\), the goal is to bring similar terms together into a single term.
The fundamental steps in simplifying involve:

• Identify the like terms present in the expression.
• Combine them by adding or subtracting their coefficients while maintaining the same variable part.
• Rewrite the simplified expression, keeping the terms clearly delineated.

In our given example, adding the coefficients 3 and 5 from the terms \(3x^3\) and \(5x^3\) gives us 8, resulting in \(8x^3\).
Simplifying expressions is a key skill in algebra that aids in solving equations and inequalities efficiently. It also reduces complexity, making it easier to carry out further operations like factoring or expanding polynomials.