Problem 23 - Elementary Algebra

Question

Factor completely. $$ -48 x+27 x 3 $$

Step-by-Step Solution

Answer

The completely factored form is \(3x(3x - 4)(3x + 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to identify the greatest common factor of the terms. The given expression is \(-48x + 27x^3\). Both terms share a factor of \(3x\).

Step 2: Factor Out the GCF

Now, factor out the GCF \(3x\) from each term of the expression:\(-48x + 27x^3 = 3x(-16 + 9x^2)\).

Step 3: Simplify the Remaining Expression

The remaining expression inside the parenthesis is \(-16 + 9x^2\). Rearrange it as a standard quadratic form:\(9x^2 - 16\).

Step 4: Recognize as a Difference of Squares

The expression \(9x^2 - 16\) is a difference of squares since it can be written as:\((3x)^2 - 4^2\).

Step 5: Factor the Difference of Squares

Apply the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), to factor \((3x)^2 - 4^2\):\(9x^2 - 16 = (3x - 4)(3x + 4)\).

Step 6: Write the Complete Factored Expression

Combine the factored GCF with the factored quadratic expression to obtain the completely factored form:\(3x(3x - 4)(3x + 4)\).

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is the largest factor shared by all terms in a polynomial. Identifying the GCF is a crucial starting point when factoring polynomials. In our original expression, -48x + 27x^3, we look for common factors in both terms. Let’s split each term:

• -48x can be factored into prime components like -1, 2, 2, 2, 2, 3, and x.
• 27x^3 can be broken down into 3, 3, 3, x, x, and x.

The common factors are 3 and x. Thus, the GCF is 3x, which we factor out. Removing this GCF simplifies the polynomial, making further factoring more straightforward. Factoring here leads to: 3x(-16 + 9x^2). By reducing original expressions to their GCF, we can simplify and eventually solve more complex algebraic problems.

Difference of Squares

The difference of squares is a special pattern in polynomial expressions. It's characterized by two perfect squares separated by a subtraction sign. Mathematically, it follows the formula:\[a^2 - b^2 = (a - b)(a + b)\]This simplifies factorization, breaking down a complex expression into simpler binomials. Let’s analyze our expression: The terms 9x^2 and 16 can be rewritten as (3x)^2 and 4^2, respectively. This matches the difference of squares pattern: \[(3x)^2 - 4^2\]Applying the difference of squares formula, our expression factors into:(3x - 4)(3x + 4).Recognizing these patterns in algebra is vital as it saves time and effort while ensuring accuracy in polynomial operations.

Quadratic Expression

A quadratic expression is a polynomial of degree 2, usually taking the form \(ax^2 + bx + c\). Understanding how to factor quadratics is an essential skill in algebra. The expression inside our parentheses, \(-16 + 9x^2\), rearranges to fit the quadratic format: \(9x^2 - 16\).Although it lacks the linear term \(bx\), it's still considered a quadratic. This particular expression doesn’t follow the standard trinomial form but belongs to the special category of a difference of squares. Therefore, it’s broken down using special rules rather than the usual factoring methods for trinomials.Recognizing different forms and patterns in quadratic expressions paves the way for efficiently solving and simplifying polynomial problems.