Problem 29 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$2 x^{3}-72 x$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial is \(2x*(x+6)*(x-6)\).

Step 1: Identify Common Factors

The first step would include identifying common factors. In this case, the common factor of the given polynomial \(2 x^{3}-72 x\) is \(2x\).

Step 2: Factor out the Greatest Common Factor

Let's factor out this greatest common factor from the polynomial. Doing so, we get: \(2x(x^{2}-36)\)

Step 3: Factor the Difference of Squares

The expression inside the parenthesis \(x^{2}-36\) is a difference of squares. This can be factored as \((x+6)(x-6)\). Substituting this back in gives the final factored form of the polynomial.

Step 4: Write the Final Factored Form

Write the final factored form of the polynomial by substituting the factored form of the difference of squares back into the expression. This gives the final answer: \(2x*(x+6)*(x-6)\).

Key Concepts

Greatest Common Factor

When factoring polynomials, the first step is often to find the Greatest Common Factor (GCF). This is the largest factor that divides each term in the polynomial. Identifying the GCF simplifies the expression, making further factoring possible.

For the polynomial \(2x^3 - 72x\), start by examining each term. The first term is \(2x^3\) and the second is \(72x\). The factors of \(2x^3\) are 2 and \(x^3\), and the factors of \(72x\) are 72 and \(x\).

• The numerical GCF of 2 and 72 is 2, since 2 is the highest number that evenly divides both.
• For the variable part, \(x^3\) and \(x\), the GCF is \(x\). This is because \(x\) is the highest power of \(x\) that appears in both terms.

Putting it all together, the GCF of the original polynomial is \(2x\). By factoring out \(2x\), you simplify the polynomial, paving the way for further factorization.

Difference of Squares

Once you've factored out the Greatest Common Factor, what's left can sometimes be further factored using the difference of squares method. A difference of squares appears in the form \(a^2 - b^2\), and it can be factored into \((a+b)(a-b)\).

In our exercise, factoring out the GCF \(2x\) from \(2x^3 - 72x\) resulted in \(2x(x^2 - 36)\). Notice that \(x^2 - 36\) is a classic example of a difference of squares:

• Here, \(x^2\) is the square of \(x\) and \(36\) is the square of 6, since \(6^2 = 36\).
• Applying the difference of squares formula, \(x^2 - 36\) can be transformed into \((x + 6)(x - 6)\).

By recognizing this pattern, you can quickly and effectively complete the factorization, simplifying complex expressions into their fully factored form.

Polynomial Factoring

Polynomial factoring is a skill used to simplify expressions and solve equations. The goal is to express the polynomial as a product of simpler polynomials. This makes solving and understanding the underlying structure of polynomial expressions easier.

For the polynomial \(2x^3 - 72x\), we executed the following steps for complete factorization:

• Identify the GCF: Found \(2x\) as the GCF and factored it out as \(2x(x^2 - 36)\).
• Difference of Squares: Recognized \(x^2 - 36\) as a difference of squares and factored it as \((x+6)(x-6)\).
• Combine Factors: Integrated these factors to express the polynomial in completely factored form: \(2x(x + 6)(x - 6)\).

These steps ensure the polynomial is broken down into its simplest components, whether you're solving for roots or simplifying expressions. Mastering these techniques gives you a powerful toolset for dealing with complex polynomial equations in algebra.