(a-2b+5c)(a-b)-(a-b-c)(2a+3c)+(6a+b)(2c-3a-5b)

Jasonbradley

2 min read

·

Jun 16, 2024

46

Share

Expanding and Simplifying the Expression: (a-2b+5c)(a-b)-(a-b-c)(2a+3c)+(6a+b)(2c-3a-5b)

This article will guide you through the process of expanding and simplifying the given algebraic expression: (a-2b+5c)(a-b)-(a-b-c)(2a+3c)+(6a+b)(2c-3a-5b).

Step 1: Expanding the Products

We begin by applying the distributive property to each set of parentheses:

• (a-2b+5c)(a-b):

  • a(a-b) - 2b(a-b) + 5c(a-b)
  • a² - ab - 2ab + 2b² + 5ac - 5bc

• (a-b-c)(2a+3c):

  • a(2a+3c) - b(2a+3c) - c(2a+3c)
  • 2a² + 3ac - 2ab - 3bc - 2ac - 3c²

• (6a+b)(2c-3a-5b):

  • 6a(2c-3a-5b) + b(2c-3a-5b)
  • 12ac - 18a² - 30ab + 2bc - 3ab - 5b²

Step 2: Combining Like Terms

Now, we group together terms with the same variables and exponents:

• a² terms: a² - 18a² = -17a²
• ab terms: -ab - 2ab - 2ab - 3ab - 30ab = -39ab
• ac terms: 5ac + 3ac + 12ac = 20ac
• b² terms: 2b² - 5b² = -3b²
• bc terms: -5bc - 3bc + 2bc = -6bc
• c² terms: -3c²

Step 3: Final Expression

Finally, we combine all the simplified terms to obtain the final simplified expression:

(a-2b+5c)(a-b)-(a-b-c)(2a+3c)+(6a+b)(2c-3a-5b) = -17a² - 39ab + 20ac - 3b² - 6bc - 3c²

Therefore, the simplified form of the given expression is -17a² - 39ab + 20ac - 3b² - 6bc - 3c².