Problem 32 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Answer: \((4a + 1)(x + y)\)

Step 1: Rearrange the terms

Rearrange the given polynomial in a way that makes it easier to identify common factors. $$ (4ax + x) + (4ay + y) $$

Step 2: Identify common factors

In each group of terms, identify the common factors - in the first group, the common factor is x, and in the second group, the common factor is y. $$ x(4a + 1) + y(4a + 1) $$

Step 3: Factor the polynomial using grouping

Now that we have the groups and common factors identified, the last step is to factor the polynomial. Notice that \((4a + 1)\) is a factor of both terms, so we can factor it out. We are left with: $$ (4a + 1)(x + y) $$ The factored polynomial is: $$ (4a + 1)(x + y) $$

Key Concepts

Grouping Method

When you encounter a polynomial that needs factoring, the grouping method can be a handy trick. This method involves organizing the polynomial into smaller segments or groups, and then factoring out the common elements in each group. Consider it similar to sorting laundry; you separate clothes into categories before washing. In our given exercise, this technique is employed by splitting the polynomial terms into two groups:

• Group 1: \(4ax + x\)
• Group 2: \(4ay + y\)

The goal of this method is to identify something common between the groups to eventually factor out. Once grouped, you look for a factor that's common within each group, which brings us to identifying and utilizing the common factors effectively. Overall, the grouping method simplifies the expression and makes it easier to extract and highlight relationships between terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent real-life situations or simply be mathematical puzzles to solve. In the exercise, we started with the expression \(4ax + x + 4ay + y\). This represents a sum of terms where each term is a product of numbers and variables.
Understanding the structure of an algebraic expression is key. Look at how terms are composed:

• Coefficient: The number part of the term, e.g., "4" in \(4ax\).
• Variable: The letter that represents an unknown number, e.g., "a" and "x".
• Constant: A term that contains no variables, although our expression here doesn’t contain an obvious constant term.

Expressions can be reorganized and manipulated to show different forms which facilitate problem-solving. The rearrangement into \((4ax + x) + (4ay + y)\) was the first step to make factoring straightforward by highlighting potential common factors.

Common Factors

Understanding common factors is crucial when factoring polynomials, especially when using the grouping method. A common factor is an expression or number that is a factor of all terms in a group. In our example, identifying common factors within each group was the bridge to solving the polynomial problem.
Once the polynomial was rearranged, each group of terms could be examined for shared factors:

• In the first group \(4ax + x\), the common factor is "x".
• In the second group \(4ay + y\), the common factor is "y".

Pulling these common factors out from each group results in highlighting "\(4a + 1\)" as a combined factor between the groups. This allowed us to factor the polynomial accurately as \((4a + 1)(x + y)\), simplifying the expression by grouping the terms and their common factors elegantly.