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Chapter 2: Problem 6

Expand and combine like terms. $$ (a+b+c)(a-b-c) $$

Short Answer

Expert verified
Question: Expand and simplify the following expression: $(a+b+c)(a-b-c)$. Answer: The expanded and simplified expression is $a^2 - b^2 - c^2 - 2bc$.

Step by step solution

01

Apply Distributive Property

First, apply the distributive property to the expression by multiplying each term in the first parentheses with each term in the second parentheses. $$ (a+b+c)(a-b-c) = a(a-b-c) + b(a-b-c) + c(a-b-c) $$
02

Expand Each Term

Now, expand each term by multiplying them with the terms inside the parentheses: $$ a(a-b-c) + b(a-b-c) + c(a-b-c) = a^2 - ab - ac + ab - b^2 - bc + ac - bc - c^2 $$
03

Combine Like Terms

Finally, combine the like terms in the expression to simplify it: $$ a^2 - ab - ac + ab - b^2 - bc + ac - bc - c^2 = a^2 - b^2 - c^2 - 2bc $$ The expanded and simplified expression is: $$ (a+b+c)(a-b-c) = a^2 - b^2 - c^2 - 2bc $$

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Key Concepts

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

Distributive Property
The distributive property is an essential tool in algebra that allows us to remove parentheses and simplify expressions by distributing one term over others. When you have an expression like \[(a+b+c)(a-b-c),\]the distributive property states that you must multiply each term inside the first set of parentheses by each term inside the second set of parentheses.
This means you handle one set of multiplications at a time:
  • Start with the first term from the first parenthesis \(a\), and multiply it by each term in the second parenthesis: \(a(a-b-c)\).
  • Then, do the same with the second term \(b\): \(b(a-b-c)\).
  • Finally, repeat for the third term \(c\): \(c(a-b-c)\).
This way, you break down the problem into smaller parts, which makes it simpler to solve. Each step brings you closer to expanding and eventually simplifying the expression.
Combining Like Terms
Once you use the distributive property and expand the expression, you'll likely end up with several terms that look similar, known as "like terms." Like terms have the same variables raised to the same power. For instance, in the expression:\[a^2 - ab - ac + ab - b^2 - bc + ac - bc - c^2,\]you can identify and group the like terms:
  • Terms like \(-ab\) and \(+ab\) can be combined.
  • Similarly, terms like\(-ac\) and \(+ac\) cancel each other, and terms like \(-bc\) appear twice, so they sum to \(-2bc\).
By combining these like terms, the expression simplifies to a cleaner state, \[a^2 - b^2 - c^2 - 2bc.\]
This streamlined process helps in reducing clutter and simplifies calculations, which is essential for solving equations efficiently.
Expanding Expressions
Expanding expressions involves multiplying the terms as indicated by the distributive property, in order to express the equation without parentheses. This process is crucial in algebra because it breaks down complex expressions to a form where all operations are clearer and terms are visible for simplification.
In the example \((a+b+c)(a-b-c),\)we multiply terms from each parenthesis:
  • Multiply \(a\) with each term: \(a \,\times\, a, \, a \,\times\, -b,\) and \(a \,\times\, -c\).
  • Do the same for \(b\) and \(c\): \(b \,\times\, a,\) \(b \,\times\, -b,\) and so forth.
After this step, you no longer have parentheses. Your expression is fully expanded with all terms laid out. This sets the stage for combining like terms, which further simplifies your expression, making it more manageable and ready for solving.

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Most popular questions from this chapter

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