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Chapter 6: Problem 71

Factor by grouping. \(a^{2}-2 a+a b-2 b\)

Short Answer

Expert verified
(a + b)(a - 2)

Step by step solution

01

Group terms

First, group the terms in pairs that can be factored. Group the terms: a虏 - 2a + ab - 2bto get a虏 - 2a + ab - 2b.
02

Factor out the greatest common factor (GCF) from each pair

Factor out the common factor from each group: a虏 - 2a becomes a(a - 2)andab - 2b becomes b(a - 2).After factoring, we get:a(a - 2) + b(a - 2).
03

Factor out the common binomial

Notice that (a - 2) is a common factor in both terms. Factor (a - 2) out: (a + b)(a - 2).

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Key Concepts

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

Grouping Terms
Let鈥檚 start with the concept of **grouping terms**. Grouping terms is a method used in factoring polynomials. In this technique, we group terms in pairs that share a common factor. This makes it easier to factor them further.
In our example, we begin with the polynomial:
a虏 - 2a + ab - 2b.
To apply the grouping method, we arrange the terms in pairs like this:
(a虏 - 2a) + (ab - 2b).
By grouping, the polynomial is arranged in a more manageable format, allowing us to spot common factors. This step is crucial because it sets up the polynomial for the next stages of factoring.
Greatest Common Factor
Next, let鈥檚 discuss the **Greatest Common Factor (GCF)**. The GCF of a polynomial is the largest factor that can divide each term in a group.
When we look at the pairs we created in our example:
(a虏 - 2a) and (ab - 2b),
we need to find the GCF for each group.
  • For (a虏 - 2a), the GCF is 'a' because 'a' is the largest factor common to both a虏 and -2a.
  • For (ab - 2b), the GCF is 'b' because 'b' is common in both ab and -2b.
Once identified, we factor out the GCF from each group:
(a虏 - 2a) becomes a(a - 2)
(ab - 2b) becomes b(a - 2)
By factoring out the GCF, we simplify the polynomial, which makes it easier to factor further in the next step.
Binomial Factoring
Lastly, let鈥檚 talk about **binomial factoring**. This step involves identifying a common binomial factor in the terms. Going back to our example:
a(a - 2) + b(a - 2),
we notice that (a - 2) is common in both terms. We can factor this out as a common binomial:
(a + b)(a - 2).
This creates a product of two binomials, which is the simplified factorization of our original polynomial. The steps of grouping terms, identifying the GCF, and binomial factoring work together to break down even complex polynomials into simpler factors that are easier to work with.

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