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Chapter 6: Problem 254

Factor completely. $$ 12 a b-6 a+10 b-5 $$

Short Answer

Expert verified
(6a + 5)(2b - 1)

Step by step solution

01

Identify and Group Like Terms

Group the expression into pairs to see if common factors can be identified in smaller parts. Here, pair the terms as follows: \( (12ab - 6a) + (10b - 5) \).
02

Factor Out the Greatest Common Factor (GCF) of Each Group

Factor out the GCF from each grouped pair. For the first group \( 12ab - 6a \), the GCF is \( 6a \). For the second group \( 10b - 5 \), the GCF is \( 5 \).\[ 6a(2b - 1) + 5(2b - 1) \]
03

Factor Out the Common Binomial

Notice each term now has a common binomial factor \( 2b - 1 \). Factor \( 2b - 1 \) out of the expression:\[ (6a + 5)(2b - 1) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. Identifying the GCF is essential in simplifying expressions and factoring polynomials.
To find the GCF:
  • List out the factors of each term.
  • Identify the common factors shared by these terms.
  • Choose the largest of these common factors.
For example, in the expression \(12ab - 6a\):
  • The factors of \(12ab\) include \(1, 2, 3, 4, 6, 12, a, b\).
  • The factors of \(6a\) include \(1, 2, 3, 6, a\).
  • The common factors are \(1, 2, 3, 6, a\) and the GCF is \(6a\).
Applying this step helps break down the polynomial for easier factoring.
Grouping Terms
Grouping terms is a strategy used to simplify the factoring process. By grouping terms strategically, you make it easier to find and factor out the GCF from each part.
For instance, consider the polynomial \(12ab - 6a + 10b - 5\). Here's how you can group these terms:
  • Group \(12ab - 6a\) together.
  • Group \(10b - 5\) together.
After grouping, the polynomial becomes \( (12ab - 6a) + (10b - 5)\). This simplification allows you to find the GCF of each group, resulting in the expression \(6a(2b-1) + 5(2b-1)\). By grouping terms effectively, you simplify the expression and make the factoring process clearer.
Binomial Factor
Once you have grouped terms and factored out the GCF, the next step involves identifying and factoring out any common binomial factors. A binomial is an algebraic expression with two terms separated by a plus or minus sign, like \( (2b-1)\).
In the expression \(6a(2b-1) + 5(2b-1)\), you can see that \(2b-1\) is present in both terms as a common factor. To factor out this common binomial factor:
  • Take \(2b-1\) as a single factor from each term.
  • Write the remaining parts of each term together in another binomial.
The resulting expression is \( (6a + 5)(2b-1)\).
Factoring out a binomial simplifies the polynomial into a product of two simpler expressions, making it easier to solve or evaluate.

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Most popular questions from this chapter

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ -11-10 x+x^{2} $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ 5 p-6+p^{2} $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ a^{2}+25 a+100 $$

In the following exercises, factor each expression using any method. $$ 6 n^{2}+5 n-4 $$

In the following exercises, factor by grouping. $$ 2 x^{2}-14 x-5 x+35 $$
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