Problem 68 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 2 c^{2}-4 c+8 c^{2}-4 c^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( -4c^{3} + 10c^{2} - 4c\).

Step 1: Identify Like Terms

Like terms are terms in an expression that have the same algebraic form. In this expression, the like terms are \(2c^{2}\) and \(8c^{2}\), \( -4c\) and \(-4c^{3}\).

Step 2: Combine Like Terms

Combine the like terms by adding or subtracting them. Do \(2c^{2} + 8c^{2} = 10c^{2}\) and \(-4c - 4c^{3}\). As these are not like terms, they cannot be combined.

Step 3: Write Final Expression

Write out the final simplified expression. The simplified expression is \(10c^{2} - 4c - 4c^{3}\). We typically write the terms in descending order of powers.

Key Concepts

Like Terms

In algebra, understanding the concept of like terms is crucial when simplifying expressions. Like terms refer to terms within an expression that have the identical variable part, meaning both the variables and their exponents match perfectly. For instance, in the expression \(2c^{2} + 8c^{2}\), the terms are like terms because both include the variable \(c\) raised to the power of 2.
When simplifying expressions, identifying like terms is the first critical step.

• Like terms simplify the equation by either adding or subtracting their coefficients while keeping the variable part unchanged.
• If terms have differing variables or powers, they cannot be combined.

Recognizing like terms lays the foundation for further manipulation of algebraic expressions, making complex expressions more manageable.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. Unlike an equation, which includes an equal sign, an algebraic expression does not assert a relationship but is rather a statement of value.
In the original problem, the expression \(2c^{2} - 4c + 8c^{2} - 4c^{3}\) is composed of multiple terms containing the variable \(c\) raised to different powers.

• An expression encapsulates a combination of terms linked through operations like addition, subtraction, multiplication, and division.
• Each term consists of a coefficient (numerical component) and a variable component that might be raised to a power.

Algebraic expressions serve as the fundamental building blocks in algebra, facilitating the representation of complex mathematical ideas in a simplified form.

Combining Like Terms

Combining like terms is a fundamental technique in algebra to simplify expressions efficiently. When you encounter an expression, the goal is to consolidate similar terms to reduce it to its simplest form. For example, with \(2c^{2} + 8c^{2} - 4c - 4c^{3}\), you identify that \(2c^{2}\) and \(8c^{2}\) are like terms because they share the same variable raised to identical powers.

• Combine by adding or subtracting their coefficients: \(2 + 8 = 10\).
• Terms like \(-4c\) and \(-4c^{3}\), due to differing exponents, cannot be combined with each other.
• Therefore, the expression simplifies to \(10c^{2} - 4c - 4c^{3}\).

Combining like terms simplifies expressions to a concise form, aiding in better clarity and ease of solving algebraic equations.