Problem 51 - Elementary and Intermediate Algebra

Question

Factor out the GCF. $$ 14 x^{2}-7 x-7 $$

Step-by-Step Solution

Answer

The GCF of the expression is 7, so factor it out to get \(7(2x^2 - x - 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

To factor out the GCF, first identify the greatest common factor of all the terms in the expression. The terms are \(14x^2\), \(-7x\), and \(-7\). The constant values are 14, 7, and 7. The GCF for these numbers is 7.

Step 2: Factor Out the GCF

Once the GCF is identified as 7, factor it out from each term in the expression. Divide each term by 7 and factor 7 out of the expression. This gives us:\[ 7(2x^2 - x - 1) \]

Step 3: Verify the Factoring

Distribute 7 back into the expression \(2x^2 - x - 1\) to ensure the original expression is obtained:\[ 7 \times 2x^2 = 14x^2, \quad 7 \times -x = -7x, \quad 7 \times -1 = -7 \]The expression checks out, confirming the factoring process is correct.

Key Concepts

Greatest Common Factor

In algebra, when trying to simplify or factor an expression, one of the first steps is to identify the Greatest Common Factor (GCF). The GCF is the largest number that divides evenly into each term in the expression.
For example, in the expression given, the terms are \(14x^2\), \(-7x\), and \(-7\). The constant values of these terms are 14, 7, and 7, respectively. The greatest common factor here is 7.
Finding the GCF is crucial because factoring it out simplifies the expression, making further algebraic operations easier. Here's a quick tip: always look for the GCF by finding the largest integer that can evenly divide the coefficients of the terms. Don't forget to check for common variables as well, though in this example, only the numerical aspect is the focus.

Polynomial Expressions

A polynomial expression is a sum of terms that consist of variables raised to different powers, multiplied by coefficients. In our example, the expression is: \(14x^2 - 7x - 7\).
Here, the polynomial has three terms:

• \(14x^2\) which is a quadratic term because it includes \(x^2\)
• \(-7x\) is a linear term because it includes \(x\)
• \(-7\) is a constant term because it does not include any variable

Understanding these components is vital when you need to factor or manipulate polynomial expressions. This concept can apply not only to simplifying polynomials but to more complex algebraic operations involving them. Each term's degree (the power of its variable) helps in organizing and simplifying these expressions.

Factoring Techniques

Factoring is a critical technique in algebra, allowing you to simplify polynomial expressions by breaking them down into products of simpler expressions.
One common technique is factoring by identifying the GCF, as demonstrated in the original exercise. Once the GCF is found (7 in this case), you divide each term by the GCF and rewrite the expression, resulting in \(7(2x^2 - x - 1)\).
Factoring techniques also extend to other methods, such as factoring trinomials, grouping, or even special cases like difference of squares. For our example, after factoring out the GCF, further techniques could be applied to factor \(2x^2 - x - 1\) if needed. Remember, choosing the appropriate technique depends on the structure of the polynomial you're dealing with. Using these techniques effectively reduces complexity and can reveal solutions to equations without having to solve them by other means.