Problem 58 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ a^{4} b^{4} c^{2} d^{2}-36 x^{2} y^{2} $$

Step-by-Step Solution

Answer

Answer: The factorization of the given binomial expression is $$(a^{4} b^{4} c^{2} d^{2} + 36x^{2}y^{2})(a^{4} b^{4} c^{2} d^{2} - 36x^{2}y^{2})$$.

Step 1: Identify the difference of squares pattern

First, observe that the given binomial has the form of $$A^2 - B^2$$, where $$A^2 = a^{4} b^{4} c^{2} d^{2}$$ and $$B^2 = 36x^{2}y^{2}$$. This indicates that we can use the difference of squares pattern to factor the expression.

Step 2: Apply the difference of squares pattern

The difference of squares pattern states that for any A and B, $$(A^2 - B^2) = (A+B)(A-B)$$. Thus, we can rewrite the given expression as: $$(a^{4} b^{4} c^{2} d^{2} + 36x^{2}y^{2})(a^{4} b^{4} c^{2} d^{2} - 36x^{2}y^{2})$$.

Step 3: Further simplify the expression

We realize that the factors \((a^{4} b^{4} c^{2} d^{2} \pm 36x^{2}y^{2})\) cannot be factored further, so the final factorization of the given binomial is: $$(a^{4} b^{4} c^{2} d^{2} + 36x^{2}y^{2})(a^{4} b^{4} c^{2} d^{2} - 36x^{2}y^{2})$$.

Key Concepts

Difference of Squares

The difference of squares is a special algebraic identity that forms the basis for factoring expressions if they fit the specific pattern. Let's break this concept down, as it is a fundamental tool in algebra.

The difference of squares happens when we have two perfect squares being subtracted. In mathematical terms, it is expressed as \(A^2 - B^2\). This pattern is significant because it can be factored neatly into the product of two binomials:

• \((A + B)(A - B)\)

This simple yet powerful formula arises from the simple distributive process, known as expanding binomials, and it helps in simplifying or breaking down complex algebraic expressions.

In the given exercise, the expression \(a^{4}b^{4}c^{2}d^{2} - 36x^{2}y^{2}\) is broken down using the difference of squares pattern. Here \(A = a^{2}b^{2}cd\) and \(B = 6xy\). Using the identity, we factor the expression as:

• \((a^{2}b^{2}cd + 6xy)(a^{2}b^{2}cd - 6xy)\)

This method of factoring is crucial in solving polynomial equations efficiently.

Binomials

Binomials are algebraic expressions that contain exactly two terms. These terms can be connected through either addition or subtraction, significantly distinguishing them from simpler terms or more complex polynomials.

Understanding binomials is essential because they frequently appear in algebra, particularly in operations like expansion and factoring. In the context of the exercise, the given equation is a binomial because it consists of two separate terms: \(a^{4}b^{4}c^{2}d^{2}\) and \(-36x^{2}y^{2}\). This separation allows us to apply specific algebraic techniques such as factoring by the difference of squares to simplify or solve the expression.

The importance of recognizing binomials lies in our ability to manipulate them to derive useful algebraic identities and solutions. When factoring binomials, we apply various methods depending on the relationship between the terms, like the difference of squares in our example.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operational symbols that form meaningful mathematical phrases. They can be as simple as a single variable or as complex as a multi-term polynomial.

In algebra, mastering how to handle these expressions by simplification, evaluation, and manipulation lies at the core of mathematical problem solving. When we engage with an algebraic expression such as \(a^{4}b^{4}c^{2}d^{2} - 36x^{2}y^{2}\), our goal is to break it down into components that are easier to work with while preserving equivalence.

Algebraic expressions become helpful tools in representing real-world problems in a simplified mathematical form. For instance, by learning how to factor expressions like the one in the exercise, we gain the ability to solve equations, study functions, and explore numerous mathematical concepts more deeply.