Problem 10 - Elementary Algebra

Question

Factor completely. $$ 6 x 4+24 x 3-2 x 2-8 x $$

Step-by-Step Solution

Answer

The expression factors as \( 2x(x+4)(3x^2-1) \).

Step 1: Group the terms

We start by rearranging the terms to make grouping easier: \( 6x^4 + 24x^3 - 2x^2 - 8x \). Notice that the first two terms have common factors, and so do the last two terms.

Step 2: Factor by grouping

Group the terms and factor out the greatest common factor from each group:\( (6x^4 + 24x^3) - (2x^2 + 8x) \).For the first group: \( 6x^4 + 24x^3 \) factor out \( 6x^3 \) to get \( 6x^3(x + 4) \).For the second group: \( -2x^2 - 8x \) factor out \( -2x \) to get \( -2x(x + 4) \).

Step 3: Factor out the common binomial factor

Notice that both groups have a common binomial \( (x + 4) \). You can factor \( (x + 4) \) out of the entire expression:\( 6x^3(x + 4) - 2x(x + 4) = (x + 4)(6x^3 - 2x) \).

Step 4: Factor the remaining polynomial

Now, simplify \( 6x^3 - 2x \) by factoring out the common factor:Factor out \( 2x \):\( 6x^3 - 2x = 2x(3x^2 - 1) \).

Step 5: Write the completely factored form

The expression \( 6x^4 + 24x^3 - 2x^2 - 8x \) is now factored completely as: \( 2x(x + 4)(3x^2 - 1) \).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is incredibly important when factoring polynomials. It refers to the largest factor that all terms in an expression share. Finding the GCF is usually the first step in simplifying algebraic expressions.
To identify the GCF, list all factors of each term. For instance, in the expression \( 6x^4 + 24x^3 \), the factors for each are:

• \( 6x^4: \) factors include \( 1, 2, 3, 6, x, x^2, x^3, x^4 \)
• \( 24x^3: \) factors include \( 1, 2, 3, 4, 6, 8, 12, 24, x, x^2, x^3 \)

Notice \( 6x^3 \) appears in both lists. It's the greatest factor they share. Therefore, you can factor \( 6x^3 \) from both terms. After removing the GCF from each term, the expression simplifies, making further steps in factoring easier.

Factor by Grouping

Factor by grouping is a handy method used to factor polynomials, especially when dealing with four terms. You group terms in pairs, then factor out the GCF from each pair. This method works particularly well when these groups share a common factor after factoring the GCF.
For the polynomial \( 6x^4 + 24x^3 - 2x^2 - 8x \), we follow these steps:

• First, group: \((6x^4 + 24x^3)\) and \((-2x^2 - 8x)\).
• Factor each group: from \( 6x^4 + 24x^3 \), factor out \( 6x^3 \) to get \( 6x^3(x + 4) \). From \(-2x^2 - 8x\), factor out \( -2x \) to get \(-2x(x + 4) \).

After factoring, you observe a common binomial factor \((x + 4)\) in both expressions, enabling the next step of factoring.

Binomial Factor

A binomial factor refers to an expression consisting of two terms that can be used to simplify polynomials further. In factor by grouping, identifying a common binomial factor is key to simplifying the polynomial.
In the expression \((6x^3(x + 4) - 2x(x + 4))\), factor by grouping has successfully highlighted \((x + 4)\) as a common binomial factor. This allows us to factor \((x + 4)\) from the entire expression. This step simplifies it to \((x + 4)(6x^3 - 2x)\). With the common binomial removed and grouped together, the polynomial is much closer to its simplest form. Recognizing and correctly factoring out the binomial factor is crucial for achieving the factored form.

Factored Form

The factored form of a polynomial is when it is expressed as a product of its factors, which are usually of lower degree than the original polynomial. It's the goal when simplifying polynomial expressions because it reveals the roots of the polynomial and simplifies calculations.
Continuing from the previous steps, once we have \((x + 4)(6x^3 - 2x)\), we focus on \(6x^3 - 2x\).
Factor \(2x\) out of \(6x^3 - 2x\):
\(6x^3 - 2x = 2x(3x^2 - 1)\).
Now, the polynomial \(6x^4 + 24x^3 - 2x^2 - 8x\) in its completely factored form becomes \(2x(x + 4)(3x^2 - 1)\). Each part of the factored expression represents a simpler component of the original polynomial. Understanding how to achieve the factored form helps in solving equations and understanding properties of the polynomial.