Problem 27 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ -2 x^{5}+128 x^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(-2x^2(x - 4)(x^2 + 4x + 16)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) of the terms in the expression \(-2x^5 + 128x^2\). The GCF is the largest expression that can divide each term. Here, both terms are divisible by \(-2x^2\).

Step 2: Factor Out the GCF

Factor out the GCF from the expression. Dividing each term by \(-2x^2\), we have:\[-2x^5 + 128x^2 = -2x^2(x^3 - 64)\].

Step 3: Recognize the Difference of Cubes

The expression inside the parentheses, \(x^3 - 64\), is a difference of cubes since \(64 = 4^3\). Recall that a difference of cubes can be factored using the formula:\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\].

Step 4: Apply the Difference of Cubes Formula

Set \(a = x\) and \(b = 4\). Plug these into the formula:\[x^3 - 4^3 = (x - 4)(x^2 + 4x + 16)\].

Step 5: Rewrite the Fully Factored Expression

Combine the factored GCF with the result from factoring the difference of cubes to get the fully factored expression:\[-2x^2(x - 4)(x^2 + 4x + 16)\].

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a crucial concept in factoring polynomials. It is the largest expression that can evenly divide each term in a polynomial expression. In the given exercise, \(-2x^5 + 128x^2\), our goal is to simplify the expression by removing common factors from each term.

Here's how you identify the GCF:

• Firstly, look at the coefficients: for \(-2\) and \(128\), the GCF is \(2\).
• Next, examine the variable parts. The terms are \(x^5\) and \(x^2\). The smallest exponent of \(x\), which is \(2\), gives us \(x^2\) as part of the GCF.
• Combine them to get the full GCF, \(-2x^2\).

Factoring out the GCF simplifies the expression significantly, making \(-2x^5 + 128x^2\) easier to work with.

Once factored out, the expression reduces to \(-2x^2(x^3 - 64)\). This process highlights the essential first step in factoring polynomials.

Difference of Cubes

The difference of cubes is a specific form of factoring that applies when you have an expression of the form \(a^3 - b^3\). Recognizing this pattern is crucial in polynomial factoring.

In our transformation of the expression \(-2x^2(x^3 - 64)\), the part \(x^3 - 64\) can be recognized as a difference of cubes, since \(64 = 4^3\).

The difference of cubes formula is: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]Here's how you apply it:

• Identify \(a\) and \(b\). For \(x^3 - 64\), \(a = x\) and \(b = 4\).
• Substitute into the formula to get \((x - 4)(x^2 + 4x + 16)\).

Understanding and applying the difference of cubes formula is a vital skill for simplifying and comprehending polynomial expressions.

Factoring Expressions

Factoring expressions is a fundamental technique in algebra used to simplify problems and solve equations. The exercise involves several key steps:

• **Identifying common factors.** This step includes recognizing the greatest common factor, which simplifies each term of your polynomial by removing common divisors.
• **Recognizing special patterns.** Patterns like the difference of cubes simplify complex expressions into more manageable parts by using specific formulas.
• **Combining results.** After factoring out the GCF and using factor formulas like the difference of cubes, we compile all parts to express the polynomial in its fully factored form.

Factoring makes it easier to work with algebraic expressions, leading to simpler solutions. The final result for our exercise, \(-2x^2(x - 4)(x^2 + 4x + 16)\), demonstrates how effective these strategies can be when combined correctly. Mastering these techniques will help in solving more complex algebraic problems efficiently.