Problem 70 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ -2 c^{3}+12 c^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(10c^3\).

Step 1: Identify Like Terms

In the expression \(-2c^{3} + 12c^{3},\)both terms involve the same variable \(c\) raised to the power of 3. Since the powers are the same (both \(c^3\)), these are like terms that can be combined.

Step 2: Combine Coefficients

To simplify, add the coefficients of the like terms together. The coefficients here are \(-2\) and \(12\). Perform the addition: \[-2 + 12 = 10.\]

Step 3: Write the Simplified Expression

Since the terms have been combined, the simplified expression is \(10c^{3}.\)This is because the power of the variable \(c\) remains unchanged when combining like terms.

Key Concepts

Combining Like Terms

Combining like terms is one of the fundamental skills needed in simplifying algebraic expressions. When you encounter an expression, the first task is identifying the "like terms." But what makes two or more terms "like terms"? In algebra, terms are considered alike if they have the same variable part, which includes having the same variable raised to the same power. This means that only the coefficients, which are the numbers in front of the variables, differ.

For instance, in the expression \(-2c^3 + 12c^3\), you can see that both terms involve \(c^3\). They are like terms because both have \(c\) raised to the power of 3. Combining these like terms involves adding or subtracting their coefficients while maintaining the variable part unchanged. This principle helps to simplify lengthy expressions to their simplest form.

Always remember:

• Focus on the variable and its power to identify like terms.
• The interaction of coefficients allows you to perform simple arithmetic operations.
• It keeps the expression neat, clear, and much easier to understand.

Coefficients

Coefficients are numbers that multiply the variable part of algebraic terms. They play a crucial role in simplifying algebraic expressions. When dealing with like terms, it's the coefficients that you combine, not the variables.

In the expression, \(-2c^3 + 12c^3\), \(-2\) and \(12\) are the coefficients. Once you identify like terms, the next step is to add or subtract these coefficients depending on the operation. In our example, performing \(-2 + 12\) results in \(10\). This new coefficient prefaces the common variable part, \(c^3\).

Key details about coefficients:

• They represent how many times the variable is included in the expression.
• Combining coefficients reduces expressions to a more manageable form.
• Significantly influences the calculation since they handle the numeric part.

Polynomial Expressions

Polynomial expressions are like a canvas for combining like terms. They are made up of multiple terms that each involve variables raised to a non-negative integer power. Understanding polynomial expressions involves recognizing the structure, which consists of terms connected by addition or subtraction.

For example, the given expression\(-2c^3 + 12c^3\), is already a simplified polynomial because it's reduced to a single term through combining like terms. In general:

• Polynomials can have any number of terms.
• Each term is a combination of constants and variables.
• They simplify to fewer terms by combining those that are alike.

Working with polynomial expressions requires identifying potential like terms across large or small expressions. Simplifying them involves collecting the like ones together via the operations on their coefficients, making calculations concise and solutions easier to interpret.