Problem 88 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$16 a^{2} x-25 y-25 x+16 a^{2} y$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(16 a^{2} x - 25 y - 25 x + 16 a^{2} y\) is \((16a^2 - 25)(x + y)\)

Step 1: Arranging the polynomial

Rearrange the polynomial in the following way: \(16 a^{2} x + 16 a^{2} y - 25 x - 25 y\)

Step 2: Grouping

Group the terms as follows: \( (16 a^{2} x + 16 a^{2} y) + (- 25 x - 25 y)\)

Step 3: Factor out common terms

Factor out common terms from each group: \(16a^2(x+y) - 25 (x+y) \)

Step 4: Factor out repeated Binomial

Notice that there is a repeated binomial, \((x + y)\). Thus we can factor out as following: \((16a^2 - 25)(x + y)\)

Key Concepts

Understanding Binomials

A binomial is a type of algebraic expression that consists of exactly two terms. These terms are often separated by a plus or a minus sign. For example, in the exercise's solution, \(x + y\) is a binomial. It has two distinct terms: \(x\) and \(y\).
Binomials are important because they can often be factored in ways that reveal further simplifications or solutions to algebraic problems. For instance, if both terms share a common factor, such as a number or a variable, it can be factored out to simplify the expression. Recognizing binomials within larger polynomial expressions will help you in not only simplifying them but also in solving equations or inequalities that involve them. In particular, binomials frequently appear as repeated factors in the factoring process, as seen in the problem's solution.

Grouping Method in Factoring

The grouping method is a factoring technique used to simplify polynomials that cannot be easily factored by standard methods like looking for common factors or employing the distributive property. This technique involves rearranging and grouping terms to make factoring simpler, as performed in the exercise.
To use the grouping method:

• First, rearrange terms to align those which may have common factors.
• Next, separate the polynomial into groups, each of which can be factored further.
• Then, factor out the greatest common factor from each group.

In the exercise, \(16 a^{2} x + 16 a^{2} y\) was grouped together because they shared a factor of \(16a^2\), and \(-25 x - 25 y\) were grouped due to a common factor of \(-25\). This method exploits the distributive law to reveal a repeated binomial that can then be factored from the entire expression.

Factoring Techniques

Factoring techniques are essential strategies in solving polynomial equations and simplifying expressions. The most common methods include:

• Finding the greatest common factor and factoring it out.
• Factoring by grouping, as shown in the original exercise.
• Using special identities like difference of squares.
• Employing the quadratic formula for second-degree polynomials.

In the provided solution, a combination of grouping and recognizing a difference of squares paved the way for factoring the expression completely. It's key to practice various factoring techniques to get comfortable with recognizing when and how they can be applied. Each technique has its specific set of scenarios where it's most effective, and often, complex problems require a combination of these methods.

Algebraic Expressions Simplification

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). Simplifying these expressions can often be accomplished by following a series of logical steps, such as combining like terms or employing factoring techniques.
In algebra, simplifying expressions is crucial because it makes them easier to work with, whether you're solving equations or plotting graphs. The goal is to rewrite an expression in its simplest or most useful form.

• Start by identifying like terms—terms that have the same variable raised to the same power—and combine them.
• Next, look for common factors among the terms and factor them out.
• Employ the use of factoring techniques to break down more complex polynomials into simpler parts.

In the context of the solved exercise, after arranging and grouping the terms, factoring was utilized to express the polynomial as a product of simpler expressions, making it much easier to handle in further calculations or applications.