Factoring algebraic expressions is a fundamental skill in mathematics. This article delves into the process of factoring the expression 9x^2 - 16y^2 - 9xz 12yz. We'll provide a step-by-step guide and explain the techniques involved, including recognizing the difference of squares and proper use of parentheses.

Introduction to Factoring Techniques

Factoring is the process of breaking down an algebraic expression into simpler factors. Proper understanding and application of factoring techniques are crucial for simplifying and solving complex algebraic equations. This guide will help you understand and apply these techniques effectively.

Step-by-Step Factorization of 9x^2 - 16y^2 - 9xz 12yz

To factor the given expression, we first separate it into two parts: the difference of squares and the remaining terms.

1. Recognizing the Difference of Squares:

The first part of the expression, 9x^2 - 16y^2, is a difference of squares. The general form of a difference of squares is a^2 - b^2 (a - b)(a b). In this case:

9x^2 - 16y^2 (3x)^2 - (4y)^2 (3x - 4y)(3x 4y)

2. Factoring the Remaining Terms:

The remaining terms are -9xz 12yz. We can factor out the common factor of -3z:

-9xz 12yz -3z(3x - 4y)

3. Combining the Factors:

Now, we can combine the factored parts to get the complete factorization:

9x^2 - 16y^2 - 9xz 12yz (3x - 4y)(3x 4y) - 3z(3x - 4y)

We notice that both terms have a common factor of (3x - 4y), so we can factor it out:

9x^2 - 16y^2 - 9xz 12yz (3x - 4y)(3x 4y - 3z)

Key Points and Common Pitfalls

Key Points:

Recognize the difference of squares and factor it accordingly. Look for common factors and factor them out. Properly use parentheses to maintain the integrity of the expression.

Common Pitfalls:

Ignoring the difference of squares can make factoring more difficult. Incorrect use of parentheses can lead to ambiguous or incorrect expressions. Forgetting to check if common factors can be further factored.

Summary and Practice

By understanding and applying the steps outlined above, you can effectively factor the given expression. Practice is essential to mastering these techniques. Here is the final factorization in a boxed format:

Final Answer:

(3x - 4y)(3x 4y - 3z)

Remember, practice makes perfect. Try factoring similar expressions to reinforce your understanding and skills in algebraic factoring.