Problem 64 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 3 x^{3}-27 x $$

Step-by-Step Solution

Answer

The completely factored form is \(3x(x - 3)(x + 3)\).

Step 1: Factor out the Greatest Common Factor (GCF)

First, find the greatest common factor (GCF) of the terms in the expression. In the expression \(3x^3 - 27x\), both terms are divisible by \(3x\). Factoring out \(3x\) gives: \(3x(x^2 - 9)\).

Step 2: Recognize the Difference of Squares

The expression inside the parentheses, \(x^2 - 9\), is a difference of squares. A difference of squares can be factored using the formula \(a^2 - b^2 = (a-b)(a+b)\). In this case, \(a = x\) and \(b = 3\), so we have \(x^2 - 9 = (x - 3)(x + 3)\).

Step 3: Write the Complete Factored Form

Replace \(x^2 - 9\) with its factored form \((x - 3)(x + 3)\) in the expression. Hence, the completely factored expression is \(3x(x - 3)(x + 3)\).

Key Concepts

Understanding the Greatest Common Factor

In mathematics, the greatest common factor (GCF) is a key tool in simplifying expressions, especially polynomials. It represents the largest number or variable that is common across all terms in an expression. To find the GCF, you need to identify the highest factors shared between terms. For example, consider the expression \(3x^3 - 27x\). Both terms share a factor of \(3x\).

Here's how it works:

• List the factors of each term. For \(3x^3\), the factors include \(3, x, x^2\), and for \(-27x\), the factors are \(3, x, 9\).
• Identify the common factors. Here, both terms share the factors \(3x\).

Once identified, you "factor out" the GCF from the expression. Thus, \(3x^3 - 27x\) becomes \(3x(x^2 - 9)\). This simplification makes subsequent steps, such as further factoring, much more straightforward.

By factoring out the GCF, you essentially reduce the complexity of the expression, making it easier to work with and further factor.

Exploring the Difference of Squares

The difference of squares is a powerful factoring pattern in algebra. It applies to expressions where two squares are subtracted, resembling the form \(a^2 - b^2\). This form can be factored into \((a-b)(a+b)\). In the expression \(x^2 - 9\), you can spot this pattern, where \(x^2\) and \(9\) are perfect squares.

Here's a breakdown:

• Recognize that \(x^2\) is \(x\) times itself, and \(9\) is \(3\) times itself, i.e., \((3)^2\).
• Since it's in the form of \(a^2 - b^2\), it can be rewritten as \((x - 3)(x + 3)\).

This transformation is critical because it splits the polynomial into simpler, linear factors.

Understanding and recognizing the difference of squares helps students factor expressions efficiently and can be applied wherever a similar pattern arises. It simplifies the process around working with polynomials and solving equations.

Writing the Expression in Factored Form

Factored form is the simplified, fully broken down version of an expression where it is expressed as a product of its factors. This is particularly handy for solving equations and simplifies mathematical computations. In the original problem, by following the steps of factoring out the GCF and recognizing the difference of squares, the expression \(3x^3 - 27x\) was rewritten as \(3x(x - 3)(x + 3)\).

Advantages of writing in factored form include:

• Simplifying the solving process by revealing roots or solutions easily.
• Reducing complex expressions into manageable calculations.
• Providing insights into the structure and properties of polynomial equations.

The completely factored form not only neatly wraps up the equation but also provides a clear view of its components and potential solutions, making it an essential skill in algebra.