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Chapter 5: Problem 27

Factor each polynomial by grouping. $$ a b+3 a+2 b+6 $$

Short Answer

Expert verified
The polynomial factors to \((b + 3)(a + 2)\).

Step by step solution

01

Group the Terms

Organize the polynomial into groups to apply factoring by grouping. The expression \( ab + 3a + 2b + 6 \) can be grouped as \((ab + 3a) + (2b + 6)\).
02

Factor Out the Greatest Common Factor in Each Group

Identify and factor out the greatest common factor from each group. From the first group \( (ab + 3a) \), the common factor is \( a \), factor it out to get \( a(b + 3) \). In the second group \( (2b + 6) \), the common factor is \( 2 \), factor it out to get \( 2(b + 3) \).
03

Factor by Grouping

Now, notice that each group contains the common binomial factor \( (b + 3) \). Factor \( (b + 3) \) out of the entire expression: \( a(b + 3) + 2(b + 3) = (b + 3)(a + 2) \). This is the factored form.

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Key Concepts

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

Factoring by Grouping
Factoring by grouping is a nifty strategy to factor polynomials that can't be easily simplified with common methods. It's often used when you have four terms in your polynomial, like in the example: \( ab + 3a + 2b + 6 \). Here's a simple way to understand it:
  • First, divide the polynomial into two groups. In our case, we grouped to form \((ab + 3a)\) and \((2b + 6)\).
  • Then, individually factor each group. This means you look for any common factors in each set of parentheses.
Identifying these groupings early on can make your process smooth and error-free. Remember, the goal is to eventually find a common binomial factor you can factor out. This will lead to a simplified form of the polynomial.
Greatest Common Factor
Finding the greatest common factor (GCF) is key when factoring, especially with the grouping method. The GCF is the largest factor shared by all terms in a group, making it the simplest number or expression that you can "pull out."
For instance, in the first group \((ab + 3a)\), the GCF is \(a\) because both terms have it in common. Factoring it out gives you \(a(b + 3)\). In the second group \((2b + 6)\), the GCF is \(2\), so factoring that out gives you \(2(b + 3)\).
  • To find the GCF, list all factors of each term.
  • Choose the largest factor common to all these terms.
Finding the GCF simplifies the expression and makes the formation of a common binomial factor鈥攁nd the final complete factorization鈥攑ossible.
Binomial Factor
The binomial factor is the common element that emerges during the grouping process. It's the part of the expression where one can factor out to perform the complete factorization of the polynomial.
After factoring out the GCF in our polynomial groups, both reduced expressions included \((b + 3)\), which is the binomial factor. This factor is important because it allows you to unify the entire polynomial under one factored form: \((b + 3)(a + 2)\).
  • The binomial factor must be identical in both grouped terms to proceed with this method.
  • It serves as the link between the separate groups, condensing them into a single expression.
Recognizing and dealing with the binomial factor properly is often the key to mastering factorization by grouping.

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Most popular questions from this chapter

Use substitution to factor each polynomial completely. See Examples 11 and 12. $$ (3 x-1)^{2}+5(3 x-1)+6 $$

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