Problem 24 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ 15 x^{2}-12 x-10 x y+8 y $$

Step-by-Step Solution

Answer

Question: Factor the given polynomial using the grouping method: \(15x^2 - 12x - 10xy + 8y\). Answer: \((3x - 2y)(5x - 4)\)

Step 1: 1. Group the terms of the given polynomial

We will group the given polynomial as follows: $$ (15x^2 - 12x) - (10xy - 8y) $$

Step 2: 2. Factor out common factors from each group

Now, we find the common factors in both groups and factor them out: $$ 3x(5x - 4) - 2y(5x - 4) $$

Step 3: 3. Factor out the common binomial

We can notice that both terms have a common binomial factor \((5x - 4)\). Let's factor it out: $$ (3x - 2y)(5x - 4) $$ So, the factored polynomial using the grouping method is: $$ (3x - 2y)(5x - 4) $$

Key Concepts

Grouping Method

The grouping method is a technique used to factor polynomials, especially those with four or more terms. It involves grouping terms into pairs, finding a common factor in each pair, and then factoring further if possible. In this exercise, we started with the polynomial \(15x^2 - 12x - 10xy + 8y\), which is a good candidate for the grouping method because it has four terms.
To apply this method, here's a friendly reminder of the steps to follow:

• Identify pairs of terms that can be grouped together.
• Factor out the greatest common factor from each pair.
• If a common binomial exists, factor it out to achieve the final factored form.

Notice that in our example, we grouped and factored the polynomial like this: \((15x^2 - 12x) - (10xy - 8y)\). This allowed us to simplify and factor further easily.

Polynomial Expressions

Polynomial expressions are mathematical phrases that involve sums and/or products of variables and coefficients. They usually appear in the form of several terms added together. A term in a polynomial is made up of a variable raised to an exponent and multiplied by a coefficient, such as \(15x^2\) or \(-10xy\).
Polynomials can have different degrees, which is the highest exponent of the variables in the expression. In our example, the polynomial \(15x^2 - 12x - 10xy + 8y\) is a second-degree polynomial because the highest power of \(x\) is 2, as seen in \(15x^2\).
Understanding the structure of polynomial expressions is crucial because it helps in determining the appropriate factoring method, such as the grouping method in our exercise. Factoring allows us to express the polynomial as a product of two binomials, making it easier to solve equations or simplify expressions.

Algebraic Factoring

Algebraic factoring is a core concept in algebra that involves writing a polynomial as a product of two or more simpler polynomials. This process simplifies the expression and often aids in solving equations. In the context of our exercise, algebraic factoring involved transforming \(15x^2 - 12x - 10xy + 8y\) into its factorized form \((3x - 2y)(5x - 4)\).
The key step in factoring is identifying common factors. In our example, we factored out \(3x\) from the first group \(15x^2 - 12x\) and \(-2y\) from the second group \(-10xy + 8y\). These steps are critical to simplify the polynomial into a format that highlights its underlying binomial structure.
Factoring polynomials isn't just about breaking down numbers; it's an essential skill that enables us to handle more complicated algebraic equations effectively. It's widely used in various applications, from simplifying expressions to solving higher-degree polynomial equations. Remember, mastering these steps can make math problems more manageable and is a vital skill in algebra!