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When working with algebraic expressions, it's crucial to understand the concept of like terms. Like terms are terms within an expression that contain the same variables raised to the same power. In simpler terms, they look almost identical except for their coefficients. In the expression given, like terms include any terms that contain the same variable parts, such as the terms with the variable raised to the third power, like \( x^{3} \). In our example, these are \( \frac{3}{4}x^{3} \) and \( \frac{1}{8}x^{3} \). Additionally, constants are also considered like terms—numbers without variables that we can add together directly. Here, the constants are \( -\frac{1}{2} \) and \( 3 \). Recognizing and combining like terms simplifies the process of working with algebraic expressions.

Simplifying a polynomial involves combining like terms to create a cleaner and more concise expression. This process reduces a polynomial to its simplest form, making it easier to work with or solve an equation.Let’s break down the simplification in our exercise:

• The first step is to identify and gather like terms such as those with the same variable and power.
• Next, we perform arithmetic operations like addition or subtraction on these like terms.
• For instance, the combination \( \frac{3}{4}x^{3} + \frac{1}{8}x^{3} \) results in \( \frac{7}{8}x^{3} \) because \( \frac{3}{4} \) is equivalent to \( \frac{6}{8} \), making it easy to see how they add to \( \frac{7}{8} \).
• Similarly, adding constants \( -\frac{1}{2} + 3 \) simplifies to \( 2.5 \).

These steps ensure the polynomial becomes more manageable, reflecting only the most necessary information.

Approaching algebra problems step-by-step is crucial for understanding and correctly simplifying expressions. This method helps break down complex processes into manageable actions. Here’s how the step-by-step solution applies to our problem:

• Step 1: Start by identifying like terms. This initial step sets up the foundation by defining what needs to be combined.
• Step 2: Proceed to add the like terms. Combining them correctly ensures the expression remains balanced and accurate.
• Step 3: Finally, combine the simplified parts into one final expression. This step gives the final answer to the problem.

Each step supports the next, showing how organized thought leads to easier problem-solving and accurate solutions.