Problem 53 - Elementary and Intermediate Algebra

Question

Factor. $$ 28 a^{3} b^{3} c+14 a^{3} c-4 b^{3} c-2 c $$

Step-by-Step Solution

Answer

The factored form is \( c(2b^3 + 1)(14a^3 - 2) \).

Step 1: Identify Common Factors

First, identify any common factors in all terms of the expression. Observe that each term contains the factor \(c\). Factor the \(c\) out of the expression.Given expression: \[ 28a^3b^3c + 14a^3c - 4b^3c - 2c \]Factoring out \(c\):\[ c(28a^3b^3 + 14a^3 - 4b^3 - 2) \]

Step 2: Group Terms

After factoring out the common factor, proceed to group the terms into pairs such that each pair has a common factor. Group them as follows:\[ (28a^3b^3 + 14a^3) + (-4b^3 - 2) \]

Step 3: Factor Each Group

Factor the greatest common factor from each grouped pair separately.For the first group, \(28a^3b^3 + 14a^3\), the common factor is \(14a^3\):\[ 14a^3(2b^3 + 1) \]For the second group, \(-4b^3 - 2\), the common factor is \(-2\):\[ -2(2b^3 + 1) \]

Step 4: Factor by Grouping

Now notice that the expression within the groups, \((2b^3 + 1)\), is identical:\[ c(14a^3(2b^3 + 1) - 2(2b^3 + 1)) \]Factor \((2b^3 + 1)\) out of both groups:\[ c((2b^3 + 1)(14a^3 - 2)) \]

Step 5: Simplify the Expression

Finally, write the completely factored expression, ensuring clarity in its simplicity:\[ c(2b^3 + 1)(14a^3 - 2) \]

Key Concepts

Understanding Common Factors

Common factors are elements that appear in multiple terms of an expression. They can be numbers, variables, or a mixture of both. In factoring, we first look for common factors because they simplify expressions, making them easier to work with.
For instance, consider the expression:

• \(28a^3b^3c + 14a^3c - 4b^3c - 2c\)

Here, each term includes the factor \(c\). Factoring \(c\) out of the expression gives us:

• \(c(28a^3b^3 + 14a^3 - 4b^3 - 2)\)

By removing the common factor \(c\), we reduce the complexity of the expression, enabling easier manipulation in subsequent steps.

Factor by Grouping

Factor by grouping is a technique used when an expression consists of four or more terms. Here, we group the terms into pairs and factor each pair separately.
Using the grouped terms from the expression \(28a^3b^3 + 14a^3c - 4b^3c - 2c\), we factor them as follows:

• Group: \((28a^3b^3 + 14a^3) + (-4b^3 - 2)\)

Next, factor each group:

• First group: The common factor is \(14a^3\) -> \(14a^3(2b^3 + 1)\)
• Second group: The common factor is \(-2\) -> \(-2(2b^3 + 1)\)

Notice that both groups contain the identical factor \(2b^3 + 1\), which simplifies to \(c((2b^3 + 1)(14a^3 - 2))\). Grouping helps to further simplify the expression.

The Greatest Common Factor: A Deeper Dive

The greatest common factor (GCF) is the largest factor shared by terms within an expression. It plays a key role in the factorization process, as it allows for simplification by "pulling out" the largest common component.
In the expression \(28a^3b^3 + 14a^3 - 4b^3 - 2\), we first identified \(c\) as a common factor across all terms. In the grouping phase, each subset of terms can be broken down further by its GCF:

• For \(28a^3b^3 + 14a^3\), the GCF is \(14a^3\).
• For \(-4b^3 - 2\), the GCF is \(-2\).

Factoring the GCF helps ensure that the expression is as simplified as possible, facilitating both calculation and understanding. It is an essential skill in recognizing and solving polynomial expressions efficiently.