Problem 64 - Algebra for College Students

Question

Factor by grouping. $$3 x^{2}+18 x-2 x-12$$

Step-by-Step Solution

Answer

Factor by grouping gives: \((3x - 2)(x + 6)\).

Step 1: Group the Terms

We start by looking at the expression: \[ 3x^2 + 18x - 2x - 12 \]We will group the terms into two pairs:\[ (3x^2 + 18x) - (2x + 12) \]

Step 2: Factor out the Greatest Common Factor (GCF) from Each Group

In the first group, \(3x^2 + 18x\), the GCF is \(3x\). In the second group, \(-2x - 12\), the GCF is \(-2\).- Factor out the GCF from the first group: \[ 3x(x + 6) \]- Factor out the GCF from the second group: \[ -2(x + 6) \]

Step 3: Combine the Factored Groups

After factoring the GCF from each group, we have:\[ 3x(x + 6) - 2(x + 6) \]Notice that \(x + 6\) is a common factor. We can factor \(x + 6\) out of the entire expression:\[ (3x - 2)(x + 6) \]

Step 4: Verify the Solution

Expand \((3x - 2)(x + 6)\) to ensure that it matches the original expression.\[ 3x(x + 6) - 2(x + 6) = 3x^2 + 18x - 2x - 12 = 3x^2 + 16x - 12 \]

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is a key concept in factoring. It represents the largest factor that can divide all terms in a given expression without leaving a remainder. Identifying the GCF is an essential step in simplifying expression terms and is particularly crucial when you are factoring polynomials.

For example, in the expression \(3x^2 + 18x\), the terms both have factors of 3 and they share the variable factor \(x\). Therefore, the GCF is \(3x\). Understanding how to find the GCF for expressions requires knowledge of basic operations:

• Identify all factors of each term.
• Find common factors present in all terms.
• Select the largest of these common factors.

Factoring out the GCF simplifies the problem into smaller, more manageable parts. This strategy is often used in conjunction with other factoring methods like factoring by grouping.

Factor by Grouping

Factoring by grouping is an effective method used to simplify polynomials, especially when working with four-term polynomials. The essence of this approach is to separate terms into groups that can each be easily factored, revealing common factors.

Here's a simplified approach:

• Group your terms: Arrange the polynomial into smaller expressions that are easier to work with. For instance, in the expression \(3x^2 + 18x - 2x - 12\), you can group it as \((3x^2 + 18x) - (2x + 12)\).
• Find the GCF for each group: Identify and factor out the GCF for each grouped term. In the first group \(3x^2 + 18x\), the GCF is \(3x\), and in the second group \(-2x - 12\), the GCF is \(-2\).
• Factor the expression: After factoring out the GCF from each group, you can combine the factored parts to simplify the expression further.

The benefit of this technique is that it simplifies complexity, as common factors are evident and can be factored out. In our example, this process results in \((3x - 2)(x + 6)\). This reflects the power and simplicity of the factor by grouping method.

Polynomial Expressions

Polynomial expressions are at the heart of algebra and involve combining variables and coefficients using operations like addition, subtraction, and multiplication. Each of these expressions consists of terms where variables are raised to whole number powers.

The hierarchy of polynomials is often determined by the degree, which is the highest power of the variable. The process of organizing and simplifying these polynomials is crucial for solving algebraic equations. When tackling polynomial expressions, it's essential to understand:

• The structure of a polynomial, which includes terms, coefficients, and exponents.
• The need for factoring to reduce expressions into more manageable forms.
• Specific methods like factoring by grouping and utilizing the GCF to simplify expressions.

Grasping these points enables students to manipulate and solve polynomial equations effectively. For instance, in the expression \(3x^2 + 18x - 2x - 12\), recognizing how terms interact and relate through factoring helps break down complex problems into simpler tasks, providing insight and understanding.