Problem 80 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$2 y^{5}-128 y^{2}$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(2 y^{5}-128 y^{2}\) is \(2y^2(y-4)(y^2+4y+16)\)

Step 1: Identify the Greatest Common Factor (GCF)

The GCF for \(2 y^{5}\) and \(-128 y^{2}\) is \(2y^2\). Factoring out the GCF gives us: \(2y^2( y^{3}-64)\)

Step 2: Recognize the Difference of Squares

Recognize that \( y^{3}-64\) is a difference of cubes. Apply the formula for the difference of cubes, where \(a^3-b^3=(a-b)(a^2+ab+b^2\). Here 'a' is 'y' and 'b' is '4'

Step 3: Apply the Difference of Cubes

Using the difference of cubes formula gives us: \(2y^2(y-4)(y^2+4y+16)\)

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, commonly abbreviated as GCF, can be thought of as the 'glue' that holds the terms of a polynomial together. It is the highest number and letter combination that can evenly divide each term within the polynomial. In other words, it is the largest expression that all terms in the polynomial share.

Imagine you have several strings of different lengths and you're trying to cut them into uniform pieces without any leftover string; the GCF would represent the longest possible length for these uniform pieces. When factoring polynomials, finding the GCF is typically our first step because it simplifies the rest of the process by reducing the polynomial to a simpler form.

How to Find the GCF

Start by listing the factors of each term and find the largest one they all have in common. This includes both the numerical coefficient (the number in front of the variable) and the variable part with its smallest exponent. For instance, if you're dealing with the terms \(2y^5\) and \(-128y^2\), both numbers are divisible by 2, and the smallest power of y they share is \(y^2\). Therefore, the GCF is \(2y^2\). Factoring out the GCF from the polynomial leads us to a simpler expression that can further be analyzed for additional factorization opportunities.

Difference of Cubes

Another key technique in the world of polynomial factorization is recognizing special patterns like the Difference of Cubes. A difference of cubes occurs when a polynomial can be expressed as the difference between two perfect cubes. In other words, it is a specific scenario where two terms are subtracted, each being a cube of another expression.

The standard formula for factorizing a difference of cubes is \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\] where both 'a' and 'b' are variables or constants.

Let's apply this to a real-world scenario: Imagine you're trying to evenly split a cube of cheese into smaller, consumable pieces, both solid and crumbled. The formula for the difference of cubes is similar to cutting that cube into a smaller cube and some crumbled pieces that represent the remaining difference.

In the polynomial \(y^3 - 64\), we would consider \(y\) to be 'a' and '4' to be 'b' (since \(64\) is \(4^3\)). By applying the difference of cubes formula, we effectively break down the expression into two simpler factors that are easier to evaluate and understand.

Polynomial Factorization Techniques

Grasping various polynomial factorization techniques is crucial for simplifying complex expressions and solving algebraic equations. These techniques are analogous to using a combination of tools to dismantle a machine to understand its parts better.

There are several techniques used in polynomial factorization, including:

• Finding the GCF for reducing the terms to their simplest form
• Identifying and applying special product rules such as the difference of squares and the sum or difference of cubes
• Factoring by grouping, which involves rearranging and grouping terms to reveal common factors
• Using the quadratic formula or factoring quadratic expressions

Each method has its place, depending on the polynomial you're dealing with. For example, with the polynomial \(2y^5 - 128y^2\), you would first identify and factor out the GCF, and then recognize that the remaining expression falls under the pattern of a difference of cubes, requiring specific strategies to break it down further. By combining these approaches, complex polynomials can be factored into products of simpler expressions, allowing for easier manipulation and solution of algebraic problems.