(a-b)(a+b)+b(a+b)-a^2

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (a-b)(a+b)+b(a+b)-a^2

This article will guide you through the process of simplifying the algebraic expression: (a-b)(a+b)+b(a+b)-a^2.

Understanding the Key Concepts

Before we dive into the simplification, let's understand the fundamental algebraic concepts involved:

• Distributive Property: This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the results. In symbols: a(b + c) = ab + ac.
• Difference of Squares: This pattern describes the factorization of the difference of two squared terms: a² - b² = (a + b)(a - b).

Simplifying the Expression

1. Expand the first two terms using the distributive property:

   (a-b)(a+b) + b(a+b) - a^2 = a(a+b) - b(a+b) + b(a+b) - a²

2. Simplify by combining like terms:

   a² + ab - ab - b² + ab + b² - a²

3. Notice the terms ab and -ab cancel out:

   a² - b² + ab + b² - a²

4. The terms a² and -a² also cancel out:

   -b² + ab + b²

5. Finally, the terms -b² and b² cancel out:

   ab

The Simplified Expression

Therefore, the simplified form of the expression (a-b)(a+b)+b(a+b)-a² is ab.

Conclusion

By applying the distributive property and recognizing the difference of squares pattern, we successfully simplified the expression. This process highlights the importance of understanding algebraic concepts and techniques to manipulate expressions efficiently.