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Chapter 5: Problem 97

Factor by grouping. \(12-4 a-3 b+a b\)

Short Answer

Expert verified
\((3 - a)(4 - b)\)

Step by step solution

01

Group Terms

Group the terms in pairs in a way that will make it easier to factor: \( (12 - 4a) + (-3b + ab) \).
02

Factor Out the Greatest Common Factor (GCF)

Factor out the greatest common factor from each pair of terms: \( 4(3 - a) - b(3 - a) \).
03

Factor the Common Binomial

Observe that \((3 - a)\) is common in both groups. Factor \((3 - a)\) out of the expression: \((3 - a)(4 - b)\).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. For example, in the exercise above, the expression is \(12-4a-3b+ab\). Each part separated by a plus or minus sign is a term.
When simplifying or factoring these expressions, it's important to understand how to manipulate the terms correctly. Always look for common factors and group the terms in a way to make the factoring easier.
Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor shared by a set of terms. In factoring, finding the GCF helps to simplify expressions by using common factors. For instance:
When we group terms from the expression \(12-4a-3b+ab\), we get \((12-4a) + (-3b+ab)\).
Next, identify the GCF in each pair:
  • For \(12-4a\): the GCF is 4. So, \(4(3 - a)\).
  • For \(-3b+ab\): the GCF is \(-b\). So, \(-b(3 - a)\).
This step simplifies the problem and makes the common binomial clearer.
Binomial Factoring
Binomial factoring involves factoring expressions composed of two terms. After grouping terms and factoring out the GCF in our problem, we are left with \(4(3 - a) - b(3 - a)\).
Notice that \(3 - a\) appears in both terms. We can factor \((3 - a)\) from each term, which simplifies the expression significantly:
Now we have \((3 - a)(4 - b)\).
This is the final factored form. Binomial factoring reduces complex expressions, making it easier to solve equations or understand their behavior.

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Most popular questions from this chapter

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