Problem 82 - Elementary Algebra

Question

Factor completely. $$ 54 x 3-2 y 3 $$

Step-by-Step Solution

Answer

\(2(3x - y)(9x^2 + 3xy + y^2)\) is the completely factored form.

Step 1: Identify Common Factor

First, let's look for the greatest common factor (GCF) of the given expression.We have the terms: \(54x^3\) and \(-2y^3\).The GCF of the coefficients \(54\) and \(2\) is \(2\). There are no common variable factors since one term is \(x^3\) and the other is \(y^3\). Hence, the GCF of the entire expression is \(2\).

Step 2: Factor out the GCF

Now, we'll factor out the GCF \(2\) from the expression. The expression can be rewritten as: \[54x^3 - 2y^3 = 2(27x^3 - y^3) \]

Step 3: Recognize Sum/Difference of Cubes Form

The remaining expression is \(27x^3 - y^3\), which is a difference of cubes. Remember that \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Step 4: Apply the Difference of Cubes Formula

Identify \(a\) and \(b\) for our expression. Here, \(a = 3x\) and \(b = y\) because \((3x)^3 = 27x^3\) and \(y^3 = y^3\).Now apply the formula: \[27x^3 - y^3 = (3x - y)((3x)^2 + 3x(y) + y^2)\] This simplifies to:\[= (3x - y)(9x^2 + 3xy + y^2)\]

Step 5: Combine the Factored Expression

The complete factored expression with the GCF included is:\[2(3x - y)(9x^2 + 3xy + y^2)\]

Key Concepts

Greatest Common Factor (GCF)

When factoring polynomials, identifying the Greatest Common Factor (GCF) is a crucial first step. It involves determining the largest factor that divides each term of the polynomial without leaving a remainder. This is important because simplifying expressions at the start makes the rest of the factoring process easier.
To find the GCF:

• List the factors of each term in the polynomial.
• Find the greatest number that is a factor of all coefficients.
• Check for common variables with the smallest exponent, if applicable.

In the expression given, we had two terms: \(54x^3\) and \(-2y^3\). Here, the coefficients are \(54\) and \(-2\). The GCF of these numbers is \(2\) because \(2\) is the largest number that divides both \(54\) and \(2\).
Thus, we factored out \(2\) from the entire expression, leading to simpler terms to work with.

Difference of Cubes

The difference of cubes is a special case in polynomial factoring. Recognizing this scenario allows us to use a specific formula for factoring. This method is applicable when you have two terms, both perfect cubes, separated by a subtraction sign.
Here’s the formula for the difference of cubes:

• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

In our exercise, the expression \(27x^3 - y^3\) fits this pattern. We identify \(a = 3x\) since \((3x)^3 = 27x^3\), and \(b = y\) because \(y^3\) is already given. Applying the formula gives us:

• \(27x^3 - y^3 = (3x - y)(9x^2 + 3xy + y^2)\)

This specific factorization is versatile and simplifies the polynomial for further calculations and insights.

Factoring Techniques

Factoring techniques help break down complex expressions into simpler factors, which are easier to manage and understand. The journey from a polynomial to its factors often includes several methods:

1. **Finding the GCF:** Always the starting point, extracting the largest factor shared by all terms.
2. **Recognizing patterns:** Patterns like the difference of cubes or sum of squares often provide shortcuts to factorization.
3. **Applying formulas:** Use known formulas to directly transform special polynomial forms, such as the one for difference of cubes.

In our example, we used the GCF to simplify into \(2(27x^3 - y^3)\) first. Then, by recognizing \(27x^3 - y^3\) as a difference of cubes, the specific formula for this pattern allowed further refinement into \((3x - y)(9x^2 + 3xy + y^2)\).
Learning and practicing these techniques not only aids in factoring but also enhances overall problem-solving skills, making polynomials less daunting and more accessible.