What Is 8b-ab+7a Subtracted From 3a-9ab+b

What is 3a - 9ab + b subtracted from 3a - 9ab + b? A Deep Dive into Algebraic Subtraction

This seemingly simple question, "What is 3a - 9ab + b subtracted from 3a - 9ab + b?", opens the door to a deeper understanding of algebraic subtraction, polynomial manipulation, and the fundamental principles of mathematical operations. While the immediate answer might seem trivial, exploring the process reveals crucial concepts applicable to more complex algebraic expressions. Let's break down this problem step-by-step and explore related concepts.

Understanding Subtraction in Algebra

In algebra, subtraction is essentially the addition of the additive inverse. The additive inverse of a number or expression is its opposite. For example, the additive inverse of 5 is -5, and the additive inverse of (3a - 9ab + b) is -(3a - 9ab + b).

Therefore, subtracting (3a - 9ab + b) from itself can be rewritten as:

(3a - 9ab + b) + (-(3a - 9ab + b))

Step-by-Step Solution

1. Rewrite the Expression: The problem states "What is 3a - 9ab + b subtracted from 3a - 9ab + b?". This translates to:

   (3a - 9ab + b) - (3a - 9ab + b)

2. Distribute the Negative Sign: The negative sign before the second parenthesis needs to be distributed to each term within the parenthesis:

   3a - 9ab + b - 3a + 9ab - b

3. Combine Like Terms: Now, we combine like terms. Like terms are terms that have the same variables raised to the same powers. In this expression, we have:

   • 3a and -3a: These are like terms. 3a - 3a = 0
   • -9ab and 9ab: These are like terms. -9ab + 9ab = 0
   • b and -b: These are like terms. b - b = 0

4. Simplify: After combining like terms, we are left with:

   0 + 0 + 0 = 0

The Result

Therefore, when you subtract (3a - 9ab + b) from (3a - 9ab + b), the result is 0.

Expanding the Concept: Subtraction of Different Polynomials

Let's now consider a more complex scenario. Suppose we want to subtract a different polynomial from (3a - 9ab + b). For example, let's subtract (2a + 5ab - 2b) from (3a - 9ab + b):

(3a - 9ab + b) - (2a + 5ab - 2b)

1. Distribute the Negative Sign:

   3a - 9ab + b - 2a - 5ab + 2b

2. Combine Like Terms:

   • 3a and -2a: 3a - 2a = a
   • -9ab and -5ab: -9ab - 5ab = -14ab
   • b and 2b: b + 2b = 3b

3. Simplify:

   a - 14ab + 3b

Therefore, (3a - 9ab + b) - (2a + 5ab - 2b) = a - 14ab + 3b

Practical Applications and Real-World Examples

The concept of subtracting algebraic expressions has widespread applications in various fields:

• Physics: Calculating net forces, where subtracting opposing forces is crucial. For instance, if one force acts in the positive direction and another in the negative direction, subtracting one from the other will provide the resultant force.

• Engineering: Determining the remaining material after a cut or removal in design and manufacturing processes.

• Finance: Calculating profit or loss by subtracting expenses from revenue. The variables could represent different types of income or expenses.

• Computer Science: In algorithm design and optimization, where subtracting values or expressions might be needed in calculations.

Advanced Concepts: Polynomials and their Properties

The expressions we've been working with are examples of polynomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Understanding polynomials is fundamental to advanced algebra and calculus.

Key Properties of Polynomials:

• Degree: The highest power of the variable in the polynomial. For example, the polynomial 3a - 9ab + b has a degree of 2 (because of the term -9ab).

• Terms: The individual parts of a polynomial separated by + or - signs.

• Coefficients: The numerical factors that multiply the variables in each term.

Troubleshooting Common Errors

When working with algebraic subtraction, several common errors can occur:

• Incorrect Distribution of the Negative Sign: Failing to distribute the negative sign to all terms within the parentheses can lead to incorrect results.

• Incorrect Combining of Like Terms: Mistakes in combining like terms often arise from overlooking the signs (+ or -) before each term.

• Ignoring Order of Operations: Remember that the order of operations (PEMDAS/BODMAS) should always be followed.

Conclusion

Subtracting algebraic expressions, even seemingly simple ones like subtracting (3a - 9ab + b) from itself, provides valuable insights into algebraic manipulation, the properties of polynomials, and the fundamental principles of mathematics. Mastering these concepts is essential for success in higher-level mathematics and its practical applications across diverse fields. By carefully following the steps outlined and understanding the concepts discussed, you can confidently tackle more complex algebraic subtraction problems. Remember to practice regularly to reinforce your understanding and improve your skills in algebraic manipulation.