Problem 71 - Elementary Algebra

Question

Factor by grouping. Do not combine like terms before factoring. $$ c^{2}-8 c-3 c+24 $$

Step-by-Step Solution

Answer

(c - 8)(c - 3)

Step 1: Identify pairs for grouping

Separate the expression into two pairs: \( c^2 - 8c \text{ and } -3c + 24 \).

Step 2: Factor out the greatest common factor (GCF) from each pair

Factor out the GCF from each group: \( c(c - 8) \text{ and } -3(c - 8) \).

Step 3: Factor out the common binomial factor

Notice that \(c - 8\) is common in both terms. Factor out \(c - 8\) from both: \( (c - 8)(c - 3) \).

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is crucial when factoring by grouping. The GCF is the largest number or term that divides evenly into each term of the polynomial. By recognizing and factoring out this common factor, we simplify the polynomials into more manageable pieces. For example, if we look at the given expression:

• The polynomial is: \( c^2 - 8c - 3c + 24 \)
• Separate the polynomial into two groups: \( (c^2 - 8c) \text{ and } (-3c + 24) \)
• In the first group, \( c^2-8c \), the GCF is \( c \). Factor it out: \( c(c - 8)\)
• In the second group, \( -3c+24 \), the GCF is \( -3 \). Factor it out: \( -3(c - 8)\)

Recognizing and factoring out the GCF allows us to reveal common terms that can be further simplified.

Binomial Factor

The binomial factor is a two-term expression that appears as a common factor in each group of the polynomial. Once you've factored out the GCF from each group, you often find that these pairs contain a common binomial factor. In our given exercise, the expression boils down to:

• Groups after factoring out the GCF: \( c(c - 8) \text{ and } -3(c - 8) \)
• You'll notice that \( c - 8 \) is a binomial factor common to both groups.
• We can then factor this common \( (c - 8) \) binomial, simplifying the expression to: \( (c - 8)(c - 3) \)

This method makes factoring simpler and provides an intuitive way to break down complex polynomials.

Factoring Polynomials

Understanding how to factor polynomials is essential in algebra. Factoring by grouping is one method used to simplify polynomials, especially when dealing with four-term polynomials. The steps involved are:

• Identify pairs within the polynomial you can group.
  Here, we paired \( c^2 - 8c \) and \( -3c + 24 \).
• Factor out the greatest common factor (GCF) from each group. This gives us \(c(c-8)\) and \(-3(c-8)\).
• Look for a common binomial factor, which is \(c-8\) in our exercise. Factor out the common binomial, resulting in: \( (c - 8)(c - 3) \).

A consistent approach to factoring polynomials such as this makes solving algebraic problems more manageable and leads to a deeper understanding of polynomial equations.