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Chapter 5: Problem 5

Remove the brackets from the given expression: \((x-1)(x+2)\)

Short Answer

Expert verified
Answer: The simplified form of the expression is \(x^2 + x - 2\).

Step by step solution

01

Distributive Property

Apply the distributive property to the expression by multiplying each term in the first parentheses set by each term in the second parentheses set. We will have: \((x-1)(x+2) = x(x+2) - 1(x+2)\)
02

Expand the Terms

Expand the terms in the expression by multiplying each term: \(x(x+2) - 1(x+2) = x^2 + 2x - x - 2\)
03

Simplify the Expression

Combine the like terms in the expression to simplify it: \(x^2 + 2x - x - 2 = x^2 + x - 2\) The final expression with the brackets removed is \(x^2 + x - 2\).

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental aspect of algebra that makes working with expressions easier. This property helps us to eliminate parentheses in expressions like \((x-1)(x+2)\). Here's how it works:
  • Multiply each term inside the first parentheses by each term inside the second parentheses.
  • In our example: \((x-1)(x+2)\) becomes \(x(x+2) - 1(x+2)\).
By applying the distributive property, we essentially distribute the terms across the expression. This process breaks down complex expressions into simpler parts, making it easier to manage as we progress towards finding a solution.
Expanding the Expressions
Once you've applied the distributive property, the next step is to expand the expressions. This involves carrying out the multiplication for each term that you've distributed:
  • For \(x(x+2)\), you'll multiply \(x\) by each term inside the parentheses: \(x \cdot x\) and \(x \cdot 2\).
  • Similarly, for \(-1(x+2)\), multiply \(-1\) by each term: \(-1 \cdot x\) and \(-1 \cdot 2\).
After expansion, our expression \(x^2 + 2x - x - 2\) represents the result of distributing and multiplying each term. Expanding helps in visualizing all individual components before simplification.
Simplifying Expressions
Simplifying expressions is the final puzzle piece in this algebraic process. Here, we look for like terms鈥攖erms that have the same variable raised to the same power鈥攁nd combine them:
  • Identify like terms in our expression: \(2x\) and \(-x\).
  • Add them together: \(2x - x\) yields \(+x\).
The simplified version of our expression is \(x^2 + x - 2\). Simplification consolidates the expression into a more concise form, making it easier to understand and use in further calculations. This step reduces complexity and enhances clarity in algebraic expressions.

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