Problem 52 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(12 x^{3}-8 x^{2}-20 x\)

Step-by-Step Solution

Answer

The complete factorization is \(4x(x + 1)(3x - 5)\).

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

Start by identifying and factoring out the greatest common factor from all terms in the expression. The GCF for the terms \(12x^3\), \(-8x^2\), and \(-20x\) is \(4x\). Thus, factor \(4x\) out of the entire expression:\[12x^3 - 8x^2 - 20x = 4x(3x^2 - 2x - 5)\]

Step 2: Identify the Quadratic Expression

Notice that inside the parenthesis we have a quadratic expression \(3x^2 - 2x - 5\). This expression will be the focus of our further factoring efforts.

Step 3: Factor the Quadratic Expression Using the AC Method

To factor the quadratic expression \(3x^2 - 2x - 5\), use the \(AC\) method. Multiply \(A=3\) and \(C=-5\), resulting in \(-15\). Find two numbers that multiply to \(-15\) and add to \(-2\), which are \(3\) and \(-5\). Hence, split the middle term accordingly:\[3x^2 + 3x - 5x - 5\]

Step 4: Grouping

Group the terms into two pairs: \((3x^2 + 3x)\) and \((-5x - 5)\). Factor out the common factor in each group:\[3x(x + 1) - 5(x + 1)\]Notice \(x + 1\) is a common factor now.

Step 5: Factor by Grouping

Since \(x + 1\) is a common factor, factor \(x + 1\) out:\[3x(x+1) - 5(x+1) = (x + 1)(3x - 5)\]

Step 6: Write the Complete Factorization

Combine the outside factor with the fully factored quadratic expression:\[12x^3 - 8x^2 - 20x = 4x(3x^2 - 2x - 5) = 4x(x + 1)(3x - 5)\]This is the complete factorization of the original expression.

Key Concepts

Greatest Common Factor (GCF)

The concept of Greatest Common Factor, or GCF, plays a crucial role in simplifying expressions. The GCF is the largest factor that is common to all terms of an expression. To find the GCF, you need to identify the highest number and the highest power of any variable that can evenly divide each term.
For the expression given, the terms are

• \(12x^3,\)
• \(-8x^2,\)
• and \(-20x\).

To find the GCF, start with the coefficients (numbers) of these terms: 12, 8, and 20. The largest number that divides 12, 8, and 20 is 4. Next, examine the variable part: \(x^3, x^2,\) and \(x\). The smallest power of \(x\) present is \(x\), because it can divide each of the other terms.

Thus, the GCF of the terms is \(4x\). By factoring \(4x\) out from the original expression, it simplifies as the crucial first step in the process of factoring completely.

Quadratic Expression

A quadratic expression is an algebraic expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. This type of expression features a squared term, making its highest exponent 2. In our context, after factoring out the GCF from the original expression, we are left with the quadratic expression \(3x^2 - 2x - 5\).

Quadratic expressions are fundamental in algebra as they appear frequently in equations and are key to solving a wide array of problems. Factoring these expressions helps find solutions to equations that would otherwise be more complex to handle. Recognizing the quadratic expression is critical because it informs the method we will use to factor it. The expression \(3x^2 - 2x - 5\) is our gateway to applying more detailed factoring techniques.

AC Method

The AC Method is a systematic way to factor quadratic expressions. It is particularly useful for quadratics where \(a\) is not equal to 1. This method involves three simple steps:

• First, calculate the product of coefficients \(A\) and \(C\). For \(3x^2 - 2x - 5\), multiply \(3\) and \(-5\), resulting in \(-15\).
• Second, find two numbers that multiply to \(-15\) (AC) and add to \(-2\) (the middle coefficient \(B\)). The numbers \(3\) and \(-5\) satisfy these conditions.
• Third, split the middle term using these numbers to create two binomials. So, \(-2x\) breaks down into \(+3x - 5x\), allowing further factoring.

Using the AC Method provides a straightforward approach to divide and conquer more complex quadratic expressions, simplifying them into products of binomials that reveal their solutions more clearly.

Factoring by Grouping

The method of factoring by grouping is an effective strategy for factoring polynomials. Once you use the AC Method to rewrite the quadratic expression, the next step involves separating terms into groups and factoring common factors out of each group.

Consider the expression after splitting: \(3x^2 + 3x - 5x - 5\).

• First group them as \((3x^2 + 3x)\) and \((-5x - 5)\).
• Within each group, factor out the greatest common factor: \(3x\) from the first and \(-5\) from the second, resulting in \(3x(x + 1)\) and \(-5(x + 1)\).

Notice \((x + 1)\) is common to both groups. Factoring out this term results in \((x + 1)(3x - 5)\).

Factoring by grouping is powerful because it reduces the expression to a simpler form, making it easier to solve or further manipulate algebraically. It relies on the symmetry of products within the expression, allowing a common term to be isolated and factored out effectively.