Problem 66 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ b c+b+c d+d $$

Step-by-Step Solution

Answer

The expression factors to \((c + 1)(b + d)\).

Step 1: Identify Common Terms

Look at the expression \( bc + b + cd + d \). Notice that the terms are grouped: \( bc + b \) and \( cd + d \). Identify that both groups of terms have a common factor. The first group, \( bc + b \), shares a common factor of \( b \). The second group, \( cd + d \), shares a common factor of \( d \).

Step 2: Factor Out the Common Factors

Factor out \( b \) from the first group which becomes \( b(c + 1) \). Factor out \( d \) from the second group which becomes \( d(c + 1) \). Now the expression becomes: \( b(c + 1) + d(c + 1) \).

Step 3: Identify and Factor Out the Common Binomial

Notice that both terms \( b(c + 1) + d(c + 1) \) have a common binomial factor of \( (c + 1) \). Factor out \( (c + 1) \) from the entire expression to get \((c + 1)(b + d)\).

Key Concepts

Understanding Common Factors

Common factors are a crucial aspect when it comes to simplifying algebraic expressions. In mathematics, a common factor of two or more numbers or terms is any number or expression that divides each of them without leaving a remainder. Identifying these factors is often the first step in the factoring process, as they can help you simplify the expression by reducing it to fewer terms.

In the given exercise, the expression is grouped as \( bc + b \) and \( cd + d \). Each of these sub-expressions shares a common factor with its terms. For instance, in the first group, \( bc + b \), both terms contain the factor \( b \). In the second group, \( cd + d \), the factor \( d \) is common.

• For \( bc + b \), identify \( b \) as the common factor.
• For \( cd + d \), identify \( d \) as the common factor.

By identifying and taking out these common factors, you simplify the expression to make it easier to factor further.

Decoding Binomial Factors

Binomial factors often come into play when a common factor doesn't simplify the expression enough. A binomial is an expression with exactly two terms, such as \( c + 1 \) in the exercise. Identifying binomial factors can seem challenging, but it simplifies the process significantly.

In the exercise, once common factors \( b \) and \( d \) are factored out, we are left with \( b(c + 1) + d(c + 1) \). Notice how each term contains a common binomial factor, \( (c + 1) \). This additional layer of factoring is crucial:

• Factor out the binomial \( (c + 1) \) from the expression.
• You're left with the expression \( (c + 1)(b + d) \).

Recognizing and factoring out common binomials allows the expression to take a simpler, factored form, streamlining further calculations or solutions.

Diving into Algebraic Expressions

Algebraic expressions are versatile tools in mathematics, consisting of constants, variables, and operations. They represent real-world quantities in simplified forms and are utilized to solve equations, inequalities, and more. The exercise we are discussing involves a basic algebraic expression with multiple variables: \( bc + b + cd + d \).

Understanding how to manipulate these expressions through factoring is vital. Factoring is essentially the reverse process of expanding; it involves breaking down an expression into simpler components (products), making it easier to work with or solve.

• Recognize repeated patterns or common elements, such as variables or products.
• Apply the appropriate factoring techniques, starting with identifying common factors, then moving to binomials or other methods as necessary.

Mastering algebraic expressions and their manipulation through methods like factoring is a building block in algebra and forms the basis for understanding more complex mathematical concepts.