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Chapter 5: Problem 96

Perform the operations. $$ \left(2 c^{2}-4 c\right)+\left(8 c-c^{2}\right) $$

Short Answer

Expert verified
The simplified expression is \( c^2 + 4c \).

Step by step solution

01

Distribute Parentheses

Since the expression already groups terms and there are no coefficients outside the parentheses, we can drop the parentheses to simplify:\( (2c^2 - 4c) + (8c - c^2) = 2c^2 - 4c + 8c - c^2 \)
02

Reorder and Combine Like Terms

Reorder and group the similar terms together:\( 2c^2 - c^2 - 4c + 8c \).Now, combine the like terms:\( (2c^2 - c^2) + (-4c + 8c) \).This simplifies to:\( c^2 + 4c \).
03

Write the Final Simplified Expression

The expression is now simplified as much as possible, with all like terms combined. The final simplified expression is \( c^2 + 4c \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
When faced with a complex expression, it's essential to simplify it to make calculations more manageable. Simplifying an expression means rewriting it in a way that is more straightforward and more conducive to further operations, without altering its value.
  • In our exercise, we started with the expression: \( (2c^2 - 4c) + (8c - c^2) \). The first step of simplification involves removing any parentheses if there are no coefficients affecting them. This case did not have coefficients interrupting our flow, allowing us to remove parentheses easily: \( 2c^2 - 4c + 8c - c^2 \).
  • The key goal is to clean up the expression by removing unnecessary grouping and combining elements where possible. This process provides clarity and makes further algebraic manipulation more straightforward.
By simplifying, we lay the groundwork for the next crucial step: identifying like terms.
Like Terms
Like terms play a fundamental role in simplifying algebraic expressions. Identifying and understanding them is crucial for efficient calculation.
  • Definition: Like terms are terms that have the same variable raised to the same power. They are the ones you can combine because they share this common characteristic.
  • For instance, in the expression \( 2c^2 - 4c + 8c - c^2 \), the terms \( 2c^2 \) and \( -c^2 \) are like terms because both involve \( c^2 \). Similarly, \( -4c \) and \( 8c \) are also like terms.
  • Recognizing like terms allows one to group and simplify these expressions by performing the operations outlined in the problem statement.
Always check and double-check like terms, as accurately combining them is the heart of simplification.
Combining Like Terms
Once like terms are identified, the next step is to combine them: a process central to simplifying algebraic expressions. This step consolidates the expression, making it easier to work with.
  • From the previous example of like terms \( 2c^2 \) and \( -c^2 \), combining them involves straightforward arithmetic: \( 2c^2 - c^2 = c^2 \).
  • Similarly, the terms \( -4c \) and \( 8c \) are combined to give \( 4c \) because \( -4 + 8 = 4 \).
  • Thus, our simplified expression becomes \( c^2 + 4c \). By using arithmetic on coefficients, like terms consolidate into a single term of their kind, reducing the overall number of terms in the expression.
Combining like terms simplifies expressions effectively, helping reveal the expression's core structure.

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Most popular questions from this chapter

Perform the indicated operations to simplify each expression, if possible. a. \(3 a+(4 a-1)+(6 a+2)\) b. \(3 a(4 a-1)(6 a+2)\)

Perform each division. $$ \frac{6 a^{2}+5 a-6}{2 a+3} $$

Perform the indicated operations to simplify each expression, if possible. a. \((4.9 a-b)-(2 a+b)\) b. \((4.9 a-b)(2 a+b)\)

Perform each division. $$ \frac{-40 f^{4}+16}{8 f^{3}} $$

Perform the indicated operations to simplify each expression, if possible. $$ \text { a. }(a+3)+\left(a^{2}-3 a+9\right) \text { b. }(a+3)\left(a^{2}-3 a+9\right) $$
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