Problem 52 - Elementary and Intermediate Algebra

Question

Factor out the GCF. $$ 27 a^{2}-9 a+9 $$

Step-by-Step Solution

Answer

The expression is factored as \(9(3a^2 - a + 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

We start by identifying the greatest common factor of the terms in the expression \(27a^2 - 9a + 9\). The terms are 27, 9, and 9 for the coefficients and we have common variable \(a\) present in the first two terms. The GCF of the numbers (27, 9, and 9) is 9 because it is the largest number that divides each of them.

Step 2: Factor Out the GCF

Now that we identified the GCF as 9, we factor it out from the expression. When we factor 9 out of \(27a^2\), we get \(3a^2\); when we factor 9 out of \(-9a\), we get \(-a\); and when we factor 9 out of \(9\), we get 1. This transforms the expression to: \[9 (3a^2 - a + 1)\].

Step 3: Review and Simplify

Review the expression \(9(3a^2 - a + 1)\) to ensure no further factoring is possible. Since there are no common factors left in the polynomial \(3a^2 - a + 1\), the factorization process is complete.

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) is fundamental in algebra, particularly in the process of factoring polynomial expressions. The GCF is the largest number that divides all the terms of a given set of numbers evenly. When factoring polynomials, you're looking for the largest factor that is common among all the terms. This step simplifies the expression by reducing it to a more manageable form.

• Identify Common Factors: Start by listing the factors of each coefficient in the polynomial. Consider the shared factors and identify the greatest one.
• Include Variables: If the polynomial includes variables, consider the lowest power of the variable common to all terms.
• Simplify the Expression: Once you have found the GCF, you can factor it out of the expression by dividing each term by this factor.

For example, in the polynomial expression from the exercise, the GCF of the coefficients 27, 9, and 9 is 9. The common factor simplifies the process of regrouping and solving. Understanding and finding the GCF is crucial, as it is the first step in breaking down complex polynomial expressions.

Coefficients

Coefficients are numbers that multiply a variable in mathematical expressions. In any polynomial expression like the one given in the exercise, such as \(27a^2 - 9a + 9\), coefficients play a significant role.

• Identify Coefficients: In the expression, the coefficients are 27, -9, and 9. These are the numerical parts that precede the variables or stand alone.
• Contribution to Factorization: Understanding coefficients is crucial because they determine the magnitude of each term and are essential for calculating the GCF.

By paying attention to coefficients, you can efficiently approach factoring. Ensuring you separate and identify these numbers helps streamline the process of solving polynomial equations. They tell us how many of each variable we're working with, and their relationship is key in simplifying expressions.

Polynomial Expression

A polynomial expression comprises multiple terms involving variables raised to whole number powers and their coefficients. It's a fundamental concept in algebra.

• Definition: A polynomial is made up of terms added together, such as \(3a^2\), \(-a\), and \(1\) in the expression \(9(3a^2 - a + 1)\).
• Factoring: The process of factoring involves breaking down the expression into simpler terms or factors. In this case, we factor out the GCF to simplify the polynomial.

Polynomials are versatile in mathematics, allowing us to model a broad range of situations. Knowing how to manipulate and factor polynomial expressions is vital for solving algebraic problems, making complex problems more approachable.