Problem 25 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 3 a+6 $$

Step-by-Step Solution

Answer

The factored expression is \(3(a + 2)\).

Step 1: Identify the Coefficients

Look at the terms in the polynomial: \(3a\) and \(6\). The coefficients of the terms are 3 and 6. Our goal is to find the greatest common factor (GCF) of these coefficients.

Step 2: Find the GCF of the Coefficients

The GCF of 3 and 6 is 3. This means the largest number that can divide both coefficients evenly is 3.

Step 3: Factor Out the GCF

Using the GCF found in Step 2, factor out 3 from each term in the polynomial. This allows us to rewrite the polynomial as: \(3(a + 2)\).

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is a fundamental concept in factoring polynomials. It is the largest number that can exactly divide each of the coefficients of the terms in a polynomial. Let's consider the polynomial given in the original exercise: \(3a + 6\). Here, the coefficients are 3 and 6.

To determine the GCF, we first list all the divisors for each coefficient. For 3, the divisors are 1 and 3, whereas for 6, the divisors are 1, 2, 3, and 6.

• Divisors of 3: 1, 3
• Divisors of 6: 1, 2, 3, 6

You can see the largest common divisor is 3. Therefore, the GCF is 3. This means 3 is the biggest number that can evenly divide both 3 and 6, making it pivotal in simplifying the polynomial.

Coefficient

Coefficients are numerical factors attached to variables in polynomial terms. Each term in a polynomial has a coefficient. In the expression \(3a + 6\), 3 and 6 are the coefficients. They are crucial because they determine how the terms behave together in operations like addition and subtraction.

When we refer to a "coefficient," we're talking about the fixed number that scales the variable. For instance, in \(3a\), 3 is the coefficient. Often in math problems, the goal is to manipulate these coefficients to simplify or transform equations.

• Changing coefficients alters the steepness or position in graph representations.
• Finding relationships between coefficients aids in understanding polynomials deeply.

Learning to accurately identify and work with coefficients is a skill that will be valuable in all sorts of mathematical scenarios, especially in factoring.

Polynomial Terms

In a polynomial, each separate part like \(3a\) or 6 is called a term. A polynomial consists of one or more terms, which are composed of a constant, a variable, or both. For the expression \(3a + 6\):

• "3a" is a term where 3 is the coefficient and \(a\) is the variable.
• "6" is a constant term since it doesn't have a variable attached to it.

Understanding polynomial terms is all about recognizing these distinct pieces. Each term is fully independent but contributes to the whole expression.

It's important to distinguish between different types of terms since it influences how we approach solving or simplifying polynomials. For any algebraic manipulation, identifying terms correctly is essential. It sets the groundwork for operations, such as finding the GCF and further factoring the polynomial.