Problem 102 - Elementary Algebra

Question

Factor out the GCF first and then factor by grouping. $$ 12 x 2-30 x-12 x+30 $$

Step-by-Step Solution

Answer

The factored form is \(6(x-1)(2x-5)\).

Step 1: Rewrite the Expression

The given expression is \[ 12x^2 - 30x - 12x + 30 \].We need to factor by grouping, which involves splitting the expression into two groups. Let's rewrite it to emphasize the grouping:\[ (12x^2 - 30x) + (-12x + 30) \].

Step 2: Factor Out the GCF from Each Group

Within the first group \((12x^2 - 30x)\), the greatest common factor (GCF) is 6x. Factoring 6x gives:\[ 6x(2x - 5) \].Within the second group \((-12x + 30)\), the GCF is 6. Factoring out 6 gives:\[ 6(-2x + 5) \].So now, the expression looks like this:\[ 6x(2x - 5) + 6(-2x + 5) \].

Step 3: Factor Out the Common Binomial

Notice that \((2x - 5)\) and \((-2x + 5)\) can be rewritten as the expression \(-(2x - 5)\). Thus, factoring out the common binomial \((2x - 5)\) gives:\[ (6x - 6)((2x - 5)) \].

Step 4: Simplify the Expression Further

We can factor a 6 out of \((6x - 6)\) to simplify more. Factoring 6 out gives:\[ 6(x - 1)(2x - 5) \].

Step 5: Final Factored Form

The final factored form of the polynomial is:\[ 6(x - 1)(2x - 5) \].

Key Concepts

Understanding the Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a key concept when dealing with polynomials. It refers to the largest factor that divides two or more numbers or terms without leaving a remainder. When factoring polynomials, finding the GCF is usually the first step.

In the given problem, the polynomial expression is \( 12x^2 - 30x - 12x + 30 \). The process begins by grouping terms and finding the GCF of each group.

Consider the first group: \( 12x^2 - 30x \).

• Identify the coefficients: 12 and 30.
• Find the GCF of the coefficients, which is 6.
• Also include the common variable factor, which is \(x\).

So, the GCF for this group is \(6x\).

Similarly, for the second group \(-12x + 30\):

• Identify coefficients: -12 and 30.
• The GCF here is also 6.

Factoring out these GCFs simplifies the polynomial step by step.

Demystifying the Binomial

A binomial is a type of polynomial that comprises exactly two terms. These two terms are often separated either by a plus sign (+) or a minus sign (-). Understanding binomials is essential in polynomial factoring.

In the expression given, we ultimately want to recognize and extract the binomials from each group. After factoring the GCF from each group, we get expressions like:

• \( 6x(2x - 5) \) from the first group
• \( 6(-2x + 5) \) from the second group

This highlights the importance of the common binomial \((2x - 5)\). Even though the second group includes \(-2x + 5\), rewriting it as \(-(2x - 5)\) standardizes the terms, allowing us to factor further. Recognizing and manipulating these binomials is crucial for successfully factoring by grouping.

Factor by Grouping Simplified

Factor by grouping is a clever technique for factoring polynomials, especially useful when a polynomial has four terms as in this problem. The process involves:

• First, group terms in pairs. For example, from \(12x^2 - 30x - 12x + 30\), create two groups: \((12x^2 - 30x)\) and \((-12x + 30)\).
• Second, factor out the GCF from each group. This showcases smaller binomials to be factored further: \(6x(2x - 5)\) and \(6(-2x + 5)\).
• Recognize a common binomial or pattern that each group shares.

Finally, extract the common binomial \((2x - 5)\) from both groups, resulting in an expression like \((6x - 6)(2x - 5)\).

To further simplify, factor any remaining GCF from the remaining terms. Here, \(6(x - 1)(2x - 5)\) is the clean, factored expression. The factor by grouping method relies on observation and practice, making it a powerful tool in algebra.