Problem 15 - A Graphical Approach to College Algebra

Question

Factor each polynomial by grouping. $$10 x^{2}-12 y+15 x-8 x y$$

Step-by-Step Solution

Answer

The factorized form is \((2x + 3)(5x - 4y)\).

Step 1: Group the Terms

First, we divide the polynomial into two groups for easier handling. Group the first two terms and the last two terms together: \( (10x^2 + 15x) + (-12y - 8xy) \).

Step 2: Factor out Common Factors

Now, factor out the greatest common factor from each group. For the first group, \(10x^2 + 15x\), the GCF is \(5x\): \( 5x(2x + 3) \).For the second group, \(-12y - 8xy\), the GCF is \(-4y\): \( -4y(3 + 2x) \).

Step 3: Rearrange the Expressions

Notice that \( (2x + 3) \) and \( (3 + 2x) \) are equivalent. Rearrange the terms to maintain uniformity:Write \(-4y(2x + 3)\) instead.

Step 4: Factor by Grouping

Both grouped terms now have a common factor of \((2x + 3)\). Factor this expression out:\((2x + 3)(5x - 4y)\).

Step 5: Verify the Factorization

Expand the factors to check correctness:\((2x + 3)(5x - 4y) = 10x^2 - 8xy + 15x - 12y\),which matches the original polynomial.

Key Concepts

Grouping Method

The grouping method is a strategic approach to factorizing polynomials, especially when you're dealing with expressions having four or more terms. This method involves rearranging terms to identify and factor out common pairs.
Let's break it down:

• **Identify Groups**: Start by looking for terms that can be bundled together. Usually, terms next to each other are grouped. For our polynomial, we grouped \((10x^2 + 15x)\) and \((-12y - 8xy)\).
• **Factor Each Group**: Within each group, check for the greatest common factors (GCF) and factor them out. Our first group gave us \(5x(2x + 3)\), while the second became \(-4y(3 + 2x)\).
• **Combine**: Check if the resulting expressions have a common binomial factor. In our case, both groups shared \((2x + 3)\), which led us to factor the polynomial into \((2x + 3)(5x - 4y)\).

Efficient factorizations make problems easier to solve or further simplify if required. Finding groups lets you visualize the structure of the polynomial better, like piecing a puzzle together.

Greatest Common Factor (GCF)

The greatest common factor (GCF) helps simplify polynomials by reducing each group to its minimal form. The GCF is the largest expression that divides each term in a group without leaving a remainder. Identifying and factoring out the GCF is a key step in the factorization process.
Here’s how it works in practice:- **Locate the GCF**: For the first group \(10x^2 + 15x\), both terms have a factor of \(5x\). Similarly, the second group \(-12y - 8xy\) has the factor \(-4y\).- **Factor Out the GCF**: Pull out these GCFs from each group. This changes the expression significantly, making it easier to see common factors across different groups.
Factoring out the GCF is useful because it often reveals simpler expressions or common factors among groups, streamlining the factorization process and making polynomials manageable.

Expanding Polynomials

After factorizing, it's always beneficial to verify by expanding the expression. This step checks if the factorization is correct and matches the original polynomial. Let's discuss expanding polynomials further.- **Multiply Out Terms**: Take the factored expression and "foil" or distribute it. For \((2x + 3)(5x - 4y)\), start by expanding each part.- **Combine Like Terms**: Group and simplify resulting terms. You should arrive back at the original polynomial. Here, \(10x^2 - 8xy + 15x - 12y\).Expanding helps reinforce your understanding of how different algebraic manipulations can both break down and rebuild polynomial expressions. It helps ensure that the factorization is accurate and the original polynomial is recaptured. Always remember that proper factorization and expansion are two sides of the same coin.