Factorise 9x 2 6x 1 25y 2

Factorising 9x² + 6x + 1 - 25y²: A Comprehensive Guide

This article provides a comprehensive guide on how to factorize the expression 9x² + 6x + 1 - 25y². We'll break down the process step-by-step, exploring different factorization techniques and highlighting key concepts in algebra. This will not only show you how to solve this specific problem, but also equip you with the skills to tackle similar algebraic expressions. We'll also touch upon the importance of recognizing patterns and the strategic use of algebraic identities.

Understanding the Expression

The expression 9x² + 6x + 1 - 25y² appears complex at first glance, but it can be elegantly factorized by strategically applying algebraic techniques. The key is to recognize that parts of this expression form a perfect square trinomial.

Perfect Square Trinomials

A perfect square trinomial is a trinomial (an expression with three terms) that can be factored into the square of a binomial. The general form of a perfect square trinomial is:

a² + 2ab + b² = (a + b)²

or

a² - 2ab + b² = (a - b)²

Let's examine our expression: 9x² + 6x + 1 - 25y²

Notice that 9x² + 6x + 1 is a perfect square trinomial. Let's see why:

• 9x²: This is (3x)²
• 1: This is (1)²
• 6x: This is 2 * (3x) * (1)

Therefore, 9x² + 6x + 1 can be factored as (3x + 1)².

Step-by-Step Factorization

Now that we've identified the perfect square trinomial, let's refactor the original expression:

1. Rewrite the expression: Replace 9x² + 6x + 1 with its factored form (3x + 1)²:

(3x + 1)² - 25y²

2. Recognize the Difference of Squares: Observe that the expression is now in the form of a difference of squares: a² - b², where a = (3x + 1) and b = 5y.

The difference of squares factorization is:

a² - b² = (a + b)(a - b)

3. Apply the Difference of Squares: Using this identity, we factorize our expression:

(3x + 1)² - (5y)² = [(3x + 1) + 5y][(3x + 1) - 5y]

4. Simplify: Finally, simplify the expression:

[(3x + 1) + 5y][(3x + 1) - 5y] = (3x + 5y + 1)(3x - 5y + 1)

Therefore, the fully factorized form of 9x² + 6x + 1 - 25y² is (3x + 5y + 1)(3x - 5y + 1).

Verifying the Factorization

To ensure our factorization is correct, we can expand the factored expression using the FOIL (First, Outer, Inner, Last) method:

(3x + 5y + 1)(3x - 5y + 1) =

• First: (3x)(3x) = 9x²
• Outer: (3x)(1) = 3x
• Inner: (1)(3x) = 3x
• Last: (1)(1) = 1
• Middle Terms with 5y: (5y)(3x) = 15xy; (-5y)(3x) = -15xy; (5y)(1) = 5y; (-5y)(1) = -5y

Combining these terms: 9x² + 3x + 3x + 1 + 15xy - 15xy + 5y - 5y = 9x² + 6x + 1

This confirms that our factorization is correct.

Expanding on Factorization Techniques

This problem highlighted the importance of recognizing specific algebraic patterns. Let's delve into some other common factorization techniques:

Greatest Common Factor (GCF)

Before attempting more complex methods, always check for a greatest common factor. This is the largest number or variable that divides evenly into all terms of the expression. If a GCF exists, factor it out first to simplify the expression.

Grouping

Grouping is a useful technique when an expression has four or more terms. You group terms with common factors, factor out the GCF from each group, and then factor out a common binomial factor.

Quadratic Trinomials

Expressions in the form ax² + bx + c can often be factored using methods like the AC method or by directly finding two numbers that add up to 'b' and multiply to 'ac'.

Cubic and Higher-Order Polynomials

Factorizing cubic and higher-order polynomials can be more challenging and often involves techniques like synthetic division or the rational root theorem.

Importance of Practice

Mastering factorization requires consistent practice. The more problems you solve, the better you'll become at recognizing patterns and selecting the appropriate technique. Start with simpler expressions and gradually work your way up to more complex ones. Don't hesitate to consult textbooks, online resources, or tutors for guidance.

Applications of Factorization

Factorization is a fundamental concept in algebra with broad applications across various fields, including:

• Calculus: Factorization is essential for simplifying expressions and solving equations in calculus.
• Physics and Engineering: Factorization is crucial for solving equations that model physical phenomena.
• Computer Science: Factorization is used in algorithms and data structures.
• Economics and Finance: Factorization helps in analyzing financial models and solving economic equations.

Conclusion

Factorizing algebraic expressions like 9x² + 6x + 1 - 25y² is a valuable skill that builds a strong foundation in algebra and higher-level mathematics. By understanding and practicing the techniques discussed in this article, you will be well-equipped to tackle similar problems and apply these skills in various contexts. Remember to always look for patterns, apply the appropriate techniques, and verify your solutions. With consistent practice and a methodical approach, you can master the art of factorization and unlock a deeper understanding of algebraic concepts.