Problem 33 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some may not be factorable. $$ x y+4 x-3 y-12 $$

Step-by-Step Solution

Answer

Question: Factor the given polynomial using the grouping method: $$xy + 4x - 3y - 12$$ Answer: The factored polynomial is $$(x - 3)(y + 4)$$

Step 1: Group terms

Group the terms with common factors. In this case, we can group "xy" and "4x" together and "-3y" and "-12" together: $$ (xy + 4x) + (-3y -12) $$ Step 2: Factor common factors from each group

Step 2: Factor common factors

For the first group, "x" is a common factor. For the second group, "3" is a common factor: $$ x(y + 4) - 3(y + 4) $$ Step 3: Factor common expression

Step 3: Factor common expression

Notice that both expressions contain "(y + 4)" as a common factor. We can factor this out: $$ (x - 3)(y + 4) $$ The polynomial has been factored using the grouping method: $$ xy + 4x - 3y - 12 = (x - 3)(y + 4) $$

Key Concepts

Grouping Method

The grouping method is a strategic approach to factoring polynomials, particularly useful when the polynomial doesn't initially appear to have any common factors across all terms. With this method, you look for terms in the polynomial that can be arranged into smaller groups where common factors are more evident. Once the polynomial is separated into these groups, you can factor out the common elements within each one.

For example, consider the polynomial \( xy + 4x - 3y - 12 \). At first glance, factoring seems tricky because there isn't a common factor across all terms. However, by rearranging the terms into two groups—\((xy + 4x)\) and \((-3y - 12)\)—you can then look within each group for common factors. This initial grouping stage is crucial and can vary by exercise, so selecting groups that reveal common factors is key to simplifying the polynomial.

Choosing the Right Groups

When using the grouping method, you may have to try arranging the terms in different ways until common factors become apparent. It's a process of trial and error that requires practice to identify the most effective grouping quickly. Once mastered, this technique becomes a powerful tool for simplifying more complex algebraic expressions.

Common Factors

Identifying common factors in algebra is fundamental to simplifying expressions and solving equations. A common factor is simply a number, variable, or combination of both that divides into two or more terms without leaving a remainder. In factoring polynomials, extracting these common factors can reduce the expression to a simpler form, which is crucial for solving algebraic problems.

In the given polynomial \( xy + 4x - 3y - 12 \), after applying the grouping method, we look within each group for common factors. The first group, \(xy + 4x\), has \(x\) as a common factor, and the second group, \((-3y - 12)\), has \(3\) as a common factor. Upon factoring these out, we get \(x(y + 4) - 3(y + 4)\).

Benefits of Factoring Out Common Factors

Factoring out common factors makes subsequent steps in solving equations more straightforward. It can also reveal patterns, such as the repetition of binomial expressions, which then can be factored further. The ability to quickly identify and extract common factors is essential for any student studying algebra.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and consist of numbers, variables, and arithmetic operations. Understanding how to manipulate these expressions is essential for solving equations, factoring polynomials, and performing a multitude of algebraic operations. When factoring, algebraic expressions can often be simplified by finding common factors or using methods such as grouping.

In our example, \( xy + 4x - 3y - 12 \), we first use the grouping method to rewrite it as two separate expressions that share a common binomial \(y + 4\). By recognizing this commonality, we simplify the expression to \(x - 3)(y + 4)\), which displays the original polynomial as a product of two simpler expressions. This factorization can then be used to solve equations or to perform further algebraic operations.

Practical Application

Mastering the manipulation of algebraic expressions is key to advancing in mathematics. The ability to factor and simplify is not just an academic exercise but also a skill that appears in practical applications, such as calculating areas, solving for unknown variables, and even in complex fields like engineering and computer science.