Factoring a^4 25b^4 - 10a^2b^2: A Step-by-Step Guide
When dealing with polynomial expressions, understanding and applying factorization techniques is crucial. This article delves into the detailed process of factoring the expression a^4 25b^4 - 10a^2b^2 using several algebraic methods and substitutions. By the end of this guide, you will have a clear understanding of how to approach similar problems.
Introduction to Factoring
Factoring, a fundamental concept in algebra and calculus, simplifies complex expressions into simpler forms. This makes them easier to analyze, compare, and manipulate. One common method is to look for patterns like perfect squares or to use substitution to reveal underlying structures in the polynomial.
Method 1: Polynomial Identification
The given expression is a^4 25b^4 - 10a^2b^2. By examining this expression, we can identify that it can be considered as a difference of squares. Specifically, it can be seen that the middle term can be rewritten to highlight this pattern.
Step-by-Step Solution
Start with the expression a^4 25b^4 - 10a^2b^2. Recognize that the middle term -10a^2b^2 is actually the square of -5ab. This can be rewritten as (-5ab)^2. Now, we can rewrite the expression as a difference of squares:a^4 25b^4 - 10a^2b^2 (a^2)^2 (5b^2)^2 - 2(a^2)(5b^2)
Further recognize this as a difference of squares, as it can be rewritten in the form (a^2 - 5b^2)^2. Thus, the factorized form of the expression is:(a^2 - 5b^2)^2
Method 2: Substitution
A more detailed algebraic method involves using substitution. Let's explore this approach step by step.
Detailed Steps
Substitute a^2 x and 5b^2 y to simplify the expression. This transforms the original expression x^2 y^2 - 2xy. Notice that x^2 y^2 - 2xy is a perfect square trinomial, and it can be factored as (x - y)^2. Substitute back x a^2 and y 5b^2 to get:(a^2 - 5b^2)^2
Method 3: Factorization by Grouping
Another approach is to group terms and factor them individually before combining them. Here's how it works:
Step-by-Step Solution
Group the terms in pairs: (a^4 - 10a^2b^2) 25b^4. Factor each pair separately: a^4 - 10a^2b^2 can be factored using the substitution x a^2 and 5y 5b^2 as (a^2 - 5b^2)(a^2 5b^2). 25b^4 is simply (5b^2)^2. Combine the factored terms:(a^2 - 5b^2)(a^2 5b^2) (5b^2)
Recognize that this can be further simplified to:(a^2 - 5b^2)(a^2 - 5b^2)
Thus, the factorized form is:(a^2 - 5b^2)^2
Conclusion
The factorization of the polynomial a^4 25b^4 - 10a^2b^2 can be achieved through different methods, including polynomial identification, substitution, and factorization by grouping. The final factorized form is (a^2 - 5b^2)^2. Understanding and practicing these methods will greatly enhance your ability to handle algebraic expressions and simplify complex polynomial factorization problems.