Problem 21 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(6 m+9\)

Step-by-Step Solution

Answer

3(2m + 3)

Step 1: Identify the Terms

The given polynomial is -6m + 9. Identify the terms of the given polynomial, which are 6m and 9.

Step 2: Find the Greatest Common Factor (GCF)

Determine the greatest common factor of the coefficients 6 and 9. The factors of 6 are 1, 2, 3, 6, and the factors of 9 are 1, 3, 9. The greatest common factor (GCF) is 3.

Step 3: Factor the GCF Out

Divide each term of the polynomial by the GCF and rewrite the polynomial as a product of the GCF and another polynomial. -6m ÷ 3 = 2m -9 ÷ 3 = 3 -So, the polynomial can be factored as 3(2m + 3).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential concept when factoring polynomials. It is the largest factor that divides two or more numbers. In simpler terms, it's the biggest number that fits into all the given terms without leaving a remainder.

To find the GCF of the coefficients in our polynomial, we need to look at the numbers alone first. Take the polynomial -6m + 9 as an example. Here, the coefficients are 6 and 9. Let’s list the factors:

• Factors of 6: 1, 2, 3, 6
• Factors of 9: 1, 3, 9

By comparing these lists, we can see that 3 is the largest number that appears in both lists. Hence, the greatest common factor (GCF) of 6 and 9 is 3. Understanding how to find the GCF helps you simplify the process of factoring polynomials.

Polynomial Terms

Polynomial terms refer to the separate parts of a polynomial that are added or subtracted together. Each term usually contains a coefficient (a number) and a variable (such as m) that may be raised to a power.

In our example, the polynomial is -6m + 9. Here, -6m is one term, and 9 is another term. Notice that:

• -6 is the coefficient of the first term, and m is the variable.
• 9 is simply a constant term, which means it doesn't contain any variable.

It is crucial to identify these terms correctly as the first step in the factoring process. They help you understand what part of the expression you will be working with when you factor out the GCF.

Factoring Process

The factoring process involves breaking down a polynomial into simpler components, usually by taking out the greatest common factor (GCF).

Here's how you do it for the polynomial \(-6m + 9\):

First, we determine the GCF, which is 3, as discussed previously. Next, we rewrite each term of the polynomial by factoring out this GCF:

• Divide each term by the GCF: -6m ÷ 3 = -2m.
• 9 ÷ 3 = 3.

Once that is done, combine these simplified terms: \(-2m + 3\). Now rewrite the original polynomial as a product of the GCF and the simplified polynomial: \3(-2m + 3)\. That's your final factored form! By following these steps, you break down the polynomial into a simpler form, making it easier to work with.

Division of Terms

Division of terms is the process employed when reworking each term by the GCF during the factoring process.

For our polynomial -6m + 9, after finding the GCF as 3, we divide each term by this GCF:

• -6m ÷ 3 results in -2m
• 9 ÷ 3 results in 3

This step is crucial as it simplifies the terms inside the polynomial. Simple division of terms allows us to reassemble them in a new structure. Once divided, these terms are then recombined: 3(-2m + 3). Hence, we see division is critical for simplification during the factoring process. By consistently dividing and recombining terms, complex polynomials become far easier to manage and solve.