Simplifying (7y² - 9y + 9) - (6y² - 9y + 7) A Step-by-Step Guide

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Polynomial expressions, a fundamental concept in algebra, often appear complex at first glance. However, with a systematic approach and a clear understanding of the underlying principles, simplifying these expressions becomes a manageable task. This comprehensive guide aims to equip you with the necessary tools and techniques to confidently simplify polynomial expressions, enhancing your algebraic proficiency. Our focal point will be the expression (7y² - 9y + 9) - (6y² - 9y + 7), which we will meticulously simplify step-by-step.

Understanding Polynomial Expressions

Before diving into the simplification process, it's crucial to grasp the basics of polynomial expressions. A polynomial expression is essentially a combination of variables (usually represented by letters like 'x' or 'y'), constants (numerical values), and mathematical operations such as addition, subtraction, and multiplication. The exponents of the variables must be non-negative integers. For example, 3x² + 2x - 5 is a polynomial expression, while 3x^(-1) + 2√x is not, due to the negative exponent and the square root (which can be expressed as a fractional exponent).

Polynomials are classified based on the number of terms they contain. A monomial has one term (e.g., 5x²), a binomial has two terms (e.g., 2x + 3), and a trinomial has three terms (e.g., x² - 4x + 7). Polynomials with more than three terms are simply referred to as polynomials. The degree of a polynomial is determined by the highest exponent of the variable in the expression. For instance, in the polynomial 4x³ - 2x² + x - 1, the degree is 3.

The Importance of Simplification

Simplifying polynomial expressions is not merely an exercise in algebraic manipulation; it's a fundamental skill that paves the way for solving more complex equations and problems. Simplified expressions are easier to work with, allowing for efficient calculations and clearer insights into the relationships between variables. In many real-world applications, polynomial expressions represent physical quantities or relationships, and simplification can lead to a more intuitive understanding of these phenomena.

Moreover, simplification is often a prerequisite for further operations, such as factoring, solving equations, and graphing functions. A simplified expression minimizes the chances of errors in subsequent steps and streamlines the problem-solving process. Therefore, mastering simplification techniques is an investment in your overall mathematical competence.

Step-by-Step Simplification of (7y² - 9y + 9) - (6y² - 9y + 7)

Now, let's embark on the simplification journey of our focal expression: (7y² - 9y + 9) - (6y² - 9y + 7). We'll break down the process into manageable steps, providing clear explanations along the way.

Step 1: Distribute the Negative Sign

The first crucial step involves distributing the negative sign in front of the second set of parentheses. This means multiplying each term inside the second parentheses by -1. This step is essential because it correctly accounts for the subtraction operation. The expression now becomes:

7y² - 9y + 9 - 6y² + 9y - 7

Notice how the signs of the terms inside the second parentheses have been flipped. The positive 6y² became negative, the negative 9y became positive, and the positive 7 became negative. This distribution is a cornerstone of simplifying expressions involving subtraction.

Step 2: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. In our expression, the like terms are: 7y² and -6y² (both have y²), -9y and +9y (both have y), and +9 and -7 (both are constants). Identifying like terms is the key to combining them and simplifying the expression.

Step 3: Combine Like Terms

This is where the actual simplification takes place. We combine the coefficients (the numerical parts) of the like terms. Let's group the like terms together for clarity:

(7y² - 6y²) + (-9y + 9y) + (9 - 7)

Now, we perform the arithmetic:

  • 7y² - 6y² = 1y² (or simply y²)
  • -9y + 9y = 0y (or simply 0)
  • 9 - 7 = 2

Step 4: Write the Simplified Expression

Putting the simplified terms together, we get:

y² + 0 + 2

Since 0 doesn't affect the sum, we can omit it, resulting in the final simplified expression:

y² + 2

Therefore, (7y² - 9y + 9) - (6y² - 9y + 7) simplifies to y² + 2. This concise form is much easier to work with in subsequent calculations or analyses.

Common Mistakes to Avoid

Simplifying polynomial expressions is a skill that improves with practice. However, certain common mistakes can hinder progress. Being aware of these pitfalls can help you avoid them and ensure accurate simplification.

  • Forgetting to Distribute the Negative Sign: This is perhaps the most frequent error. When subtracting a polynomial expression, it's crucial to distribute the negative sign to every term inside the parentheses. Failing to do so will lead to an incorrect result.
  • Combining Unlike Terms: Only like terms can be combined. Terms with different variables or the same variable raised to different powers cannot be added or subtracted. For example, 3x² and 2x are unlike terms and cannot be combined.
  • Sign Errors: Pay close attention to the signs of the terms, especially when combining like terms. A simple sign error can drastically change the final answer. Double-check your work to ensure accuracy.
  • Incorrect Order of Operations: Remember the order of operations (PEMDAS/BODMAS). Exponents should be handled before multiplication and division, which come before addition and subtraction. Applying the correct order is essential for accurate simplification.

Practice Problems

To solidify your understanding and hone your simplification skills, try working through these practice problems:

  1. (4x² + 3x - 2) - (2x² - x + 5)
  2. (5a³ - 2a + 1) + (3a³ + 4a² - 6)
  3. (9b² - 7b + 4) - (b² + 7b - 3)

By tackling these problems, you'll gain confidence in your ability to simplify polynomial expressions effectively.

Conclusion

Simplifying polynomial expressions is a fundamental skill in algebra that empowers you to solve more complex problems and gain a deeper understanding of mathematical relationships. By mastering the techniques outlined in this guide, including distributing the negative sign, identifying like terms, combining like terms, and avoiding common mistakes, you can confidently simplify a wide range of polynomial expressions. Remember, practice is key to proficiency, so keep working at it, and you'll soon find simplification becomes second nature. The simplified form y² + 2 of our initial expression, (7y² - 9y + 9) - (6y² - 9y + 7), showcases the power of these techniques in action. Embrace the challenge, and watch your algebraic abilities flourish!