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(a+b-c)(a+b+c)

3 min read Jun 16, 2024
(a+b-c)(a+b+c)

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

The expression (a + b - c)(a + b + c) represents the product of two trinomials. To expand this, we can use the distributive property or the FOIL method.

Using the Distributive Property

The distributive property states that for any numbers a, b, and c: a(b + c) = ab + ac

We can apply this property to our expression:

  1. Distribute the first trinomial over the second: (a + b - c)(a + b + c) = a(a + b + c) + b(a + b + c) - c(a + b + c)

  2. Distribute each term of the first trinomial: = (a * a + a * b + a * c) + (b * a + b * b + b * c) - (c * a + c * b + c * c)

  3. Simplify by multiplying: = a² + ab + ac + ba + b² + bc - ca - cb - c²

  4. Combine like terms: = a² + b² - c² + 2ab

Using the FOIL Method

The FOIL method is a mnemonic for remembering the order of multiplications when expanding two binomials. It stands for First, Outer, Inner, Last. While we are working with trinomials here, we can apply the same principle:

  1. Multiply the first terms of each trinomial: a * a = a²

  2. Multiply the outer terms of each trinomial: a * c = ac

  3. Multiply the inner terms of each trinomial: b * a = ba

  4. Multiply the last terms of each trinomial: -c * c = -c²

  5. Repeat steps 1-4 for each term in the first trinomial:

    • b * a = ba
    • b * b = b²
    • b * c = bc
    • -c * a = -ca
    • -c * b = -cb
    • -c * c = -c²

  6. Combine all the terms: = a² + ac + ba - c² + ba + b² + bc - ca - cb - c²

  7. Combine like terms: = a² + b² - c² + 2ab

Conclusion

Both methods lead to the same simplified expression: a² + b² - c² + 2ab. This expression is a trinomial with three terms. It can also be seen as a special case of the difference of squares pattern, where (a + b)² - c² = (a + b + c)(a + b - c).

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