Factoring Completely A²b² + 2ab + 1 A Step By Step Guide
In this comprehensive guide, we will delve into the process of factoring the algebraic expression a²b² + 2ab + 1 completely. Factoring is a fundamental skill in algebra, enabling us to simplify expressions, solve equations, and gain deeper insights into the relationships between variables. This article provides a step-by-step approach to factoring this expression, along with explanations and relevant concepts to solidify your understanding. We will begin by recognizing the pattern within the expression, then apply the appropriate factoring technique to arrive at the final, factored form. By mastering this skill, you will be well-equipped to tackle more complex algebraic problems.
To effectively factor the expression a²b² + 2ab + 1, we first need to recognize its structure. Key to this is identifying patterns that match known algebraic identities. Upon closer inspection, we can observe that the expression consists of three terms: a²b², 2ab, and 1. The first term, a²b², is the square of ab. The third term, 1, is the square of 1. The middle term, 2ab, is twice the product of ab and 1. This arrangement strongly suggests that the expression might be a perfect square trinomial.
A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is x² + 2xy + y² or x² - 2xy + y². These trinomials can be factored as (x + y)² and (x - y)², respectively. Recognizing this pattern is crucial for efficient factoring. By comparing our given expression a²b² + 2ab + 1 with the general form x² + 2xy + y², we can see a clear correspondence. We can consider ab as x and 1 as y. The middle term, 2ab, then fits the 2xy pattern perfectly, as it is twice the product of ab and 1. This recognition allows us to confidently apply the perfect square trinomial factoring technique.
As mentioned, the ability to recognize patterns is paramount in algebra, especially when factoring. In our expression, a²b² + 2ab + 1, the pattern that stands out is that of a perfect square trinomial. A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. There are two forms of perfect square trinomials:
- (x + y)² = x² + 2xy + y²
- (x - y)² = x² - 2xy + y²
Our given expression, a²b² + 2ab + 1, closely resembles the first form. To solidify this understanding, let's break down the expression:
- The first term, a²b², is the square of ab. We can write this as (ab)².
- The last term, 1, is the square of 1. We can write this as 1².
- The middle term, 2ab, is twice the product of ab and 1. This can be written as 2(ab)(1).
Therefore, we can rewrite the expression as (ab)² + 2(ab)(1) + 1². Comparing this with the general form x² + 2xy + y², we can see a direct correspondence:
- x corresponds to ab
- y corresponds to 1
This confirms that our expression is indeed a perfect square trinomial. Recognizing this pattern simplifies the factoring process significantly, as we can directly apply the formula (x + y)² to factor the expression. This recognition step is often the key to efficiently solving factoring problems and requires a keen eye for algebraic structures.
Now that we have identified the expression a²b² + 2ab + 1 as a perfect square trinomial, we can apply the appropriate factoring formula. As established, the expression fits the pattern x² + 2xy + y², where x = ab and y = 1. The factoring formula for this pattern is:
x² + 2xy + y² = (x + y)²
Substituting ab for x and 1 for y in the formula, we get:
(ab)² + 2(ab)(1) + 1² = (ab + 1)²
This direct substitution allows us to factor the expression efficiently. The result, (ab + 1)², is the factored form of the given expression. It represents the square of the binomial (ab + 1). To further illustrate this, we can expand (ab + 1)² using the binomial expansion formula or by simply multiplying (ab + 1) by itself:
(ab + 1)² = (ab + 1)(ab + 1)
Using the distributive property (also known as the FOIL method), we multiply each term in the first binomial by each term in the second binomial:
(ab + 1)(ab + 1) = (ab)(ab) + (ab)(1) + (1)(ab) + (1)(1)
Simplifying this, we get:
= a²b² + ab + ab + 1
Combining like terms, we arrive back at our original expression:
= a²b² + 2ab + 1
This confirms that our factoring is correct. Applying the perfect square trinomial formula directly is a powerful technique that simplifies the factoring process. By recognizing the pattern and using the formula, we can efficiently factor expressions like a²b² + 2ab + 1.
After applying the perfect square trinomial formula to the expression a²b² + 2ab + 1, we have arrived at the factored form:
(ab + 1)²
This result signifies that the original trinomial can be expressed as the square of the binomial (ab + 1). The factored form is crucial for various algebraic manipulations, such as simplifying expressions, solving equations, and analyzing functions. It provides a more compact and manageable representation of the expression. Understanding the factored form allows us to see the structure of the expression more clearly and can reveal important properties. For instance, we can easily identify the roots of the expression if it were set equal to zero. In this case, (ab + 1)² = 0 implies that ab + 1 = 0, which can be further solved for a or b depending on the context.
Furthermore, the factored form is useful in calculus when finding derivatives and integrals of more complex functions. By factoring a polynomial, we can often simplify the differentiation or integration process. In general, factoring is a fundamental skill in mathematics that is applied across various branches of the subject. Mastering this skill provides a solid foundation for more advanced topics. The factored form (ab + 1)² not only represents the solution to our factoring problem but also opens up avenues for further mathematical exploration and application.
In conclusion, we have successfully factored the expression a²b² + 2ab + 1 completely. The key to this process was recognizing the perfect square trinomial pattern. By identifying this pattern, we were able to apply the formula x² + 2xy + y² = (x + y)² directly, with x = ab and y = 1. This led us to the factored form:
(ab + 1)²
This result demonstrates the power of pattern recognition in algebra. Recognizing algebraic patterns like the perfect square trinomial can significantly simplify the factoring process. It is crucial to practice identifying these patterns to become proficient in factoring. Factoring is a fundamental skill in mathematics, and mastering it will greatly aid in solving more complex problems in algebra, calculus, and other areas. The ability to factor expressions allows us to simplify equations, find roots, and gain a deeper understanding of mathematical relationships. In this case, factoring a²b² + 2ab + 1 into (ab + 1)² provides a concise and useful representation of the expression.
This exercise underscores the importance of a systematic approach to factoring. First, we analyzed the expression to identify its structure. Then, we recognized the perfect square trinomial pattern. Finally, we applied the appropriate formula to arrive at the factored form. This step-by-step approach can be applied to other factoring problems as well. By consistently practicing and applying these techniques, you can build confidence and proficiency in factoring algebraic expressions. The factored form (ab + 1)² is not just the answer to this specific problem but also a testament to the power of algebraic manipulation and pattern recognition.
Final Answer: The final answer is (C)
