Problem 61 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$8 b^{2}+20 b c+2 b c^{2}+5 c^{3}$$

Step-by-Step Solution

Answer

The given expression can be factored by grouping as follows: Divide the expression into two groups and factor out the GCF from each group. Then, factor out common binomial expression. The factored form of the expression is \( (2b + 5c)(4b + c^2) \).

Step 1: Identify Groups

Divide the expression into two groups: \( (8b^2 + 20bc) + (2bc^2 + 5c^3) \)

Step 2: Factor out GCF from each group

Now factor out the greatest common factor from each group: \( 4b(2b + 5c) + c^2(2b + 5c) \)

Step 3: Factor out common binomial expression

Since both terms have a common binomial expression \((2b + 5c)\), we can factor it out: \( (2b + 5c)(4b + c^2) \) The expression is now factored by grouping. The final factored form is:

Step 4: Final Answer

\( (2b + 5c)(4b + c^2) \)

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept in factoring, especially when dealing with polynomials. The GCF of two or more numbers or terms is the largest number or term that divides them all without leaving a remainder. Identifying the GCF is often the first step in simplifying an expression or factoring by grouping.

To determine the GCF of a group of terms:

• List all the factors of each term.
• Identify the common factors among these terms.
• Select the largest factor that appears in each of the terms' lists.

In the expression \(8b^2 + 20bc\), the GCF is \(4b\), and for \(2bc^2 + 5c^3\), the GCF is \(c^2\). This makes it easier to factor each group smoothly, setting the stage for further simplification.

Binomial Expression

A binomial expression is a polynomial expression that consists of exactly two terms. Binomials are essential in various algebraic operations, including addition, subtraction, and especially factoring.

In the context of factoring by grouping, recognizing a binomial pattern is crucial. For example, the groups from the expression \( (8b^2 + 20bc) + (2bc^2 + 5c^3) \) are factored to reveal a common binomial expression \((2b + 5c)\).

• This common binomial simplifies the factoring process by allowing us to extract it from each group.
• The method depends on identifying these common terms, which is a key element of understanding polynomial behavior.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components, or factors, that when multiplied together give the original polynomial. This process can make polynomial expressions easier to work with, especially in solving equations.

There are various methods for factoring polynomials, including:

• Factoring by grouping
• Using special product formulas
• Factoring trinomials

For the given expression, factoring by grouping is employed. Here, the polynomial is divided into smaller groups, and the GCF of each group is factored out. This results in a simplified expression where further common factors, such as a binomial, can be identified and extracted.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (additions, subtractions, etc.). Understanding algebraic expressions is fundamental to many algebra applications.

With expressions such as \(8b^2 + 20bc + 2bc^2 + 5c^3\), we work through:

• Identifying different terms
• Recognizing patterns and structures
• Applying mathematical operations to simplify them

In the context of the exercise, the algebraic expression undergoes a transformation through the method of factoring by grouping. Recognizing how terms can be rearranged and grouped based on their factors is crucial to reaching the factored expression \((2b + 5c)(4b + c^2)\). This transformation not only simplifies the expression but also helps in solving equations where the expression appears.