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Chapter 5: Problem 145

Factor each polynomial completely. $$w^{2}-2 w+3 a w-6 a$$

Short Answer

Expert verified
Factor completely: \( (w-2)(w + 3a) \)

Step by step solution

01

Group Terms

Group the terms of the polynomial to make them easier to factor. Group: \( (w^{2} - 2w) + (3aw - 6a) \)
02

Factor Out the Greatest Common Factor (GCF)

Factor out the greatest common factor from each group. From the first group, factor out \(w\): \( w(w - 2) \). From the second group, factor out \(3a\): \( 3a(w - 2) \).
03

Factor by Grouping

Now, notice that \( (w - 2) \) is a common factor in both groups. Factor out \( (w - 2) \) from both terms: \( w(w - 2) + 3a(w - 2) = (w-2)(w + 3a) \).

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Key Concepts

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

Grouping Terms
When you have a polynomial with multiple terms, it can help to rearrange them into smaller groups. This makes it easier to spot common factors. In our example, the polynomial is: 
w^{2}-2w+3aw-6a
We want to group these terms in such a way that makes it easier to factor. Usually, you want each group to have something in common. Here, we'll group: 
(w^{2}-2w) + (3aw-6a)
Now, each group is ready for the next step. Grouping terms this way helps to break down the problem into smaller, more manageable pieces.
Greatest Common Factor
The greatest common factor (GCF) is the largest factor that can divide two or more numbers or terms without leaving a remainder. In the context of polynomials, finding the GCF of a group of terms involves identifying the highest degree of each variable and the largest number that divides all coefficients.

From the first group 
(w^{2}-2w)
the GCF is 
w
, so we factor it out:
w(w-2)
In the second group 
(3aw-6a)
, the GCF is 
3a
, so we factor it out: 
3a(w-2)


Factoring the GCF simplifies each group and prepares them for further factoring steps.
Factor by Grouping
After factoring out the GCF from each group, your next goal is to look for common elements between the resulting groups. This is called 'factoring by grouping'. In our current polynomial, we have:
w(w-2) + 3a(w-2)

Both groups contain 
(w-2)
as a factor. We can factor this common part out, simplifying the expression further:
(w-2)(w + 3a)

Now, the polynomial is completely factored! Factoring by grouping is a powerful tool for handling polynomials with multiple variables and coefficients effectively by finding and factoring out common elements.

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