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Chapter 5: Problem 46

Write an equivalent expression by factoring. $$a c+a d+b c+b d$$

Short Answer

Expert verified
(a + b )(c + d )

Step by step solution

01

- Group Common Factors

Identify and group the terms with common factors together. In this case, we can group it as follows: a c + a d + b c + b d .
02

- Factor out the Common Factors in Each Group

Factor out the common factor from each group: a (c + d) + b (c + d).
03

- Combine Like Terms

Notice that the c+d term is common in both resulting expressions. Factor it out: (a + b )(c + d )

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Key Concepts

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

Common Factors
To start factoring an expression, it's essential to identify the **common factors**. These are values or variables that appear in multiple terms of the expression and can be factored out and simplified. In the given exercise, each term is analyzed for common factors. For instance, in the expression \( ac + ad + bc + bd \), both the terms \( a \) and \( b \) can be grouped based on their common factors with \( c \) and \( d \). Identifying common factors is key to simplifying and reorganizing the expression.
Grouping Terms
After identifying common factors, the next step is **grouping terms** with similar factors together. In our example, the expression \( ac + ad + bc + bd \) can be grouped into pairs: \( ac + ad \) and \( bc + bd \). Here, we're reorganizing terms to make the factoring process clearer. Each group now shares a common factor that can be factored out in the next step. This method simplifies the expression and prepares it for further factorization.
Equivalent Expressions
By factoring out the common factors from each group, we can transform the original expression into an **equivalent expression**. For instance, after grouping, we factor out \( a \) from \( ac + ad \) and \( b \) from \( bc + bd \), resulting in \( a(c + d) + b(c + d) \). Though the expression looks different, it retains the same value for all variable inputs. Ensuring that the new expression is equivalent to the original confirms the accuracy of our factoring.
Algebraic Manipulation
The final step is performing **algebraic manipulation** by combining like terms. Notice in our example, the term \( c + d \) is common in both \( a(c + d) \) and \( b(c + d) \). We factor \( c + d \) out to get \((a + b)(c + d)\). This step simplifies the expression further. Algebraic manipulation involves using arithmetic operations and various algebraic rules to simplify expressions or solve equations.

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Most popular questions from this chapter

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