Problem 96 - Introductory Algebra for College Students

Question

Factor completely: \(12 x^{2}+14 x-6\) (Section 6.5, Example 2)

Step-by-Step Solution

Answer

The factorized form of the given trinomial \(12 x^{2}+14 x-6\) is \(2(2x+3)(3x-1)\)

Step 1: Finding the Common Factors

The coefficients in this trinomial don't share common factors except 1, but the task is to factor completely, so simply factor out the common factor, which in this case is 2, to make the trinomial simpler. This leaves us with \(2(6x^{2}+7x-3)\).

Step 2: Creating a Useful Pair of Numbers

The aim is to find two numbers that multiply to \(-18(6*-3)\) and add to 7 (the coefficient of x). This pair of numbers is 9 and -2 because \(9*-2 = -18\) and \(9-2 = 7\).

Step 3: Rewriting Middle Term

Rewrite the middle term of the trinomial using the pair of numbers found. The trinomial becomes \(2(6x^{2}+9x-2x-3)\).

Step 4: Factoring

Factor by grouping. This makes the equation become \(2[3x(2x+3)-(2x+3)]\), which simplifies to \(2(2x+3)(3x-1)\).

Key Concepts

Trinomial

A trinomial is a polynomial with exactly three terms. In the expression \(12x^2 + 14x - 6\), you can see three parts: \(12x^2\), \(14x\), and \(-6\). Each of these is a separate term:

- \(12x^2\): The term with \(x\) squared.
- \(14x\): The term with \(x\).
- \(-6\): The constant term.

Understanding the structure of a trinomial is crucial for factoring. When factoring trinomials, the goal is often to break them down into simpler binomials (polynomials with two terms). This process helps simplify solving equations or finding roots more easily.

Common Factor

Finding a common factor is an essential step in factoring polynomials. A common factor is a number or expression that divides each term of the polynomial without leaving a remainder. In the trinomial \(12x^2 + 14x - 6\), the first step is to look for any common factors among the coefficients 12, 14, and 6.

Here, these coefficients don't have a shared factor other than 1, but if we ignore the negative signs and factor out 2, we simplify the expression significantly. This turns the trinomial into \(2(6x^2 + 7x - 3)\), which is much easier to work with.

By factoring out the greatest common factor, you help reduce the complexity of the polynomial, making subsequent factoring steps more straightforward.

Factor by Grouping

Factor by grouping is a method used to simplify polynomials by grouping terms together and finding common factors in these groups. Let's break it down using our example. Starting with \(2(6x^2 + 7x - 3)\), the goal is to rewrite it so that it can be grouped effectively.

1. **Look for a Pair:** Find two numbers that multiply to the product of the first and last coefficients (\(-18\) in this case) and add to the middle coefficient (7). These numbers are 9 and -2.
2. **Rewrite the Expression:** Replace the middle term \(7x\) with \(9x - 2x\) to create a grouping of terms: \(2(6x^2 + 9x - 2x - 3)\).
3. **Group and Factor:** Group terms to factor common factors within them:
- Group 1: \(3x(2x + 3)\)
- Group 2: \(-1(2x + 3)\)
This allows us to factor out \((2x + 3)\), resulting in \(2(2x+3)(3x-1)\).

This method is particularly useful when the trinomial doesn't easily split into two binomials, allowing for an organized approach to factoring.