(a+b+c)(a−b−c)

Jasonbradley

2 min read

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Jun 16, 2024

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Expanding (a+b+c)(a−b−c)

This expression represents the product of two binomials, one with a positive sign before b and c and the other with a negative sign before them. To expand it, we can use the distributive property of multiplication.

Let's break it down:

1. Distribute the first term (a) of the first binomial:

   • a * (a - b - c) = a² - ab - ac

2. Distribute the second term (b) of the first binomial:

   • b * (a - b - c) = ab - b² - bc

3. Distribute the third term (c) of the first binomial:

   • c * (a - b - c) = ac - bc - c²

4. Combine all the terms:

   • (a² - ab - ac) + (ab - b² - bc) + (ac - bc - c²)

5. Simplify by combining like terms:

   • a² - b² - c² - 2bc

Therefore, the expanded form of (a + b + c)(a - b - c) is a² - b² - c² - 2bc.

Note:

• This result is similar to the difference of squares pattern, where (x + y)(x - y) = x² - y².
• The expansion highlights a specific pattern of terms when multiplying binomials with a combination of positive and negative signs.