Problem 68 - Algebra A Combined Function

Question

Factor. $$ 54 y^{3}-128 $$

Step-by-Step Solution

Answer

The expression \(54y^3 - 128\) factors to \(2(3y - 4)(9y^2 + 12y + 16)\).

Step 1: Identify the Form of the Expression

The given expression is \(54y^3 - 128\). Notice it is a difference of terms, where one term involves a cube of a variable. However, it is not a simple difference of cubes, so we'll start by considering common factors.

Step 2: Find the Greatest Common Factor (GCF)

The coefficients 54 and 128 both have the greatest common factor of 2. Thus, the GCF of the entire expression is 2. We can factor this out from the expression.

Step 3: Factor Out the GCF

By factoring out 2, the expression becomes \[54y^3 - 128 = 2(27y^3 - 64)\]. Now, observe the remaining expression, \(27y^3 - 64\).

Step 4: Recognize the Difference of Cubes

The expression \(27y^3 - 64\) is a difference of cubes. It can be written as \((3y)^3 - 4^3\). Recall the difference of cubes formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).

Step 5: Apply the Difference of Cubes Formula

Using the formula with \(a = 3y\) and \(b = 4\), apply the difference of cubes formula: \[(3y)^3 - 4^3 = (3y - 4)((3y)^2 + 3y \cdot 4 + 4^2)\].

Step 6: Simplify the Factored Expression

Simplify the expression within the parentheses: \((3y - 4)(9y^2 + 12y + 16)\).

Step 7: Write the Complete Factored Form

Combine the GCF factored out in step 3 with the result from step 6: \[2(3y - 4)(9y^2 + 12y + 16)\]. This is the fully factored form of the expression.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an essential building block in the world of algebra, especially when you need to simplify or factor an expression. The GCF is the largest number that evenly divides all the terms in an expression. In this context, we are looking at the expression \(54y^3 - 128\).
First, focus on finding the GCF of the coefficients (54 and 128) alone. Begin by identifying all the factors of each number.

• 54 factors into 1, 2, 3, 6, 9, 18, 27, and 54.
• 128 factors into 1, 2, 4, 8, 16, 32, 64, and 128.

The largest common number in these lists is 2, which tells us that the GCF for the expression is 2. Factoring out the GCF means you express the original expression in terms of this factor. This reduces the expression to a simpler form, making subsequent steps much easier.

Difference of Cubes

In algebra, recognizing special patterns such as the difference of cubes can help simplify expressions efficiently. The difference of cubes involves expressions of the form \(a^3 - b^3\), which can be factored using a specific formula: \((a-b)(a^2 + ab + b^2)\).
Once you've factored out the GCF from \(54y^3 - 128\) to get \(2(27y^3 - 64)\), you can then apply the difference of cubes formula. Here, \(27y^3\) can be expressed as \((3y)^3\) and \(64\) as \(4^3\), making the expression \((3y)^3 - 4^3\). This fits the difference of cubes pattern perfectly.
By applying the formula:
\((3y - 4)((3y)^2 + 3y \cdot 4 + 4^2)\)
you simplify the expression even further.

Factoring Techniques

Factoring techniques are strategies used to simplify algebraic expressions. Let's dive into these techniques by dissecting \(54y^3 - 128\).
Firstly, factor out the GCF, which we identified as 2, resulting in \(2(27y^3 - 64)\).
The next step involves recognizing any familiar patterns. In our case, the remaining expression \(27y^3 - 64\) is identified as a difference of cubes.
Using the difference of cubes formula:

• Identify your \(a\) as \(3y\) and \(b\) as 4.
• Apply the formula to express it as \((3y - 4)(9y^2 + 12y + 16)\).

Combine these steps to write the fully factored form as \(2(3y - 4)(9y^2 + 12y + 16)\). This combination of factoring out the GCF and using the difference of cubes sets the foundation for solving complex algebraic expressions.