(a-b)(a+b)(a-3b)

Jasonbradley

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Jun 16, 2024

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Expanding the Expression (a-b)(a+b)(a-3b)

This article will guide you through expanding the given expression: (a-b)(a+b)(a-3b). This involves using the distributive property and recognizing the special product patterns.

Understanding the Special Product Patterns

Before we start, let's recall two important special product patterns:

1. Difference of Squares: (x + y)(x - y) = x² - y²
2. Square of a Binomial: (x + y)² = x² + 2xy + y²

Expanding the Expression

1. Focus on the First Two Factors:

   • Notice that (a-b)(a+b) is in the form of the difference of squares.
   • Applying the pattern, we get: (a-b)(a+b) = a² - b²

2. Multiply the Result with the Third Factor:

   • Now, we have (a² - b²)(a-3b)
   • Use the distributive property to expand:
     • a² * (a-3b) - b² * (a-3b)
     • a³ - 3a²b - ab² + 3b³

Final Result

The fully expanded form of the expression (a-b)(a+b)(a-3b) is a³ - 3a²b - ab² + 3b³.

Key Points

• Recognizing and applying special product patterns can significantly simplify the expansion process.
• The distributive property is essential for multiplying polynomials.
• Practice with different examples to become comfortable with these concepts.