Problem 37 - Introductory Algebra for College Students

Question

Factor: \(6 x^{3}-6 x^{2}-120 x\)

Step-by-Step Solution

Answer

The factored form of the expression \(6 x^{3}-6 x^{2}-120 x\) is \(6x(x-5)(x+4)\).

Step 1: Identify the GCF

The GCF is the greatest factor that divides all the terms. In the expression \(6 x^{3}-6 x^{2}-120 x\), each term has a factor of \(6x\). So, \(6x\) is the GCF.

Step 2: Factor out the GCF

Divide each term in the expression by the GCF. This gives us \(6 x^{3} ÷ 6x = x^{2}\), \(6 x^{2} ÷ 6x = x\), and \(-120 x ÷ 6x = -20\). So the expression becomes \(6x\(x^{2}-x-20\)\).

Step 3: Factorize the quadratic

Factorize the quadratic \(x^{2}-x-20\). This can be factored into \((x-5)(x+4)\). Thus, the completely factored expression is \(6x(x-5)(x+4)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest factor that evenly divides each term in an expression. It's like finding the largest building block they all share. In the polynomial expression \(6x^3 - 6x^2 - 120x\), the task is to identify this largest shared factor.
To find the GCF, examine the coefficients and variables present in each term:

• Coefficients: 6
• Variables: Each term has at least one factor of \(x\)

For our example, 6 is the largest number that divides 6, 6, and 120 evenly. In terms of the variable, \(x\) is common to all terms, each with at least one "\(x\)" factor.
Thus, the GCF here is \(6x\), which is factored out from the expression. This step simplifies the polynomial and reveals a simpler expression within the brackets.

Quadratic Factoring

Quadratic factoring involves breaking down a quadratic expression into two binomials. Once the GCF is factored out, the problem \(6x(x^2 - x - 20)\) requires factoring the quadratic \(x^2 - x - 20\).
Here’s how to factor a quadratic:

• Look for two numbers that multiply to the constant term (-20) and add up to the coefficient of the linear term (-1).
• The numbers -5 and 4 meet these criteria: \((-5) \times 4 = -20\) and \((-5) + 4 = -1\).
• Hence, the quadratic expression can be rewritten as \((x - 5)(x + 4)\).

By practicing this method, quadratic equations become easier to manage, allowing you to express them as products of binomials.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. The example \(6x^3 - 6x^2 - 120x\) illustrates how complex these can initially appear. But by applying techniques like finding the GCF and quadratic factoring, they become more manageable.
This expression includes:

• Polynomial Terms: Terms combined through addition or subtraction.
• Coefficients: Numerical factors (6, -6, -120).
• Variables: Symbols representing numbers, typically "x" in this case.

Processing algebraic expressions involves transforming and simplifying them for further analysis or solving. Understanding each part and knowing how to manipulate them through factoring allows for clear solutions and deeper insight into their structure. This exercise shows how initially daunting expressions can unravel into clear, solvable parts.