Factor The Polynomial $6x + 15 + 6x^2$ By Finding The Greatest Common Monomial Factor And Rewrite The Expression.

Jun 19, 2025 by ADMIN 114 views

Factoring Polynomials with the Greatest Common Monomial Factor

Factoring polynomials is a fundamental skill in algebra. It's the reverse process of expanding expressions, and it's essential for solving equations, simplifying expressions, and understanding the behavior of functions. One of the first and most important factoring techniques is finding the greatest common monomial factor (GCMF). This article will guide you through the process of identifying the GCMF, factoring it out of a polynomial, and rewriting the expression in a simplified form. We will use the example polynomial $6x + 15 + 6x^2$ to illustrate the steps.

Understanding the Greatest Common Monomial Factor

The greatest common monomial factor (GCMF) is the largest monomial (a term with a numerical coefficient and variables raised to non-negative integer powers) that divides evenly into each term of the polynomial. In simpler terms, it's the biggest factor that all the terms in the polynomial share. Finding the GCMF involves two main steps:

Finding the greatest common factor (GCF) of the coefficients (the numerical parts of the terms).
Identifying the variables with the lowest exponent that are common to all terms.

Before we dive into the example, let's break down these steps further.

Finding the Greatest Common Factor (GCF) of Coefficients

The greatest common factor (GCF) of a set of numbers is the largest number that divides evenly into all of them. There are a few methods to find the GCF, but one common method is to list the factors of each number and identify the largest factor they share.

For example, let's say we want to find the GCF of 12, 18, and 30.

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

The common factors are 1, 2, 3, and 6. The greatest common factor is 6.

Identifying Common Variables with the Lowest Exponent

When dealing with polynomials, we also need to consider the variable parts of the terms. A variable is common to all terms only if it appears in every term. When identifying the GCMF, we take the common variables with the lowest exponent present in any of the terms. This ensures that the GCMF will divide evenly into each term.

For instance, consider the polynomial $x^3y^2 + x^2y^4 + x^5y$ . The variables x and y are present in all three terms.

The lowest exponent of x is 2 (in the term $x^2y^4$ ).
The lowest exponent of y is 1 (in the term $x^5y$ ).

Therefore, the variable part of the GCMF would be $x^2y$ .

Factoring $6x + 15 + 6x^2$ : A Step-by-Step Guide

Now, let's apply these concepts to our example polynomial: $6x + 15 + 6x^2$ .

Step 1: Rearrange the Terms (Optional but Recommended)

It's often helpful to rearrange the terms in descending order of their exponents. This makes it easier to identify the GCMF and write the factored expression in a standard form. So, we rewrite the polynomial as:

$6x^2 + 6x + 15$

Step 2: Find the GCF of the Coefficients

Our coefficients are 6, 6, and 15. Let's find their GCF:

Factors of 6: 1, 2, 3, 6
Factors of 15: 1, 3, 5, 15

The common factors are 1 and 3. The greatest common factor is 3.

Step 3: Identify Common Variables

Now, let's look at the variable parts of the terms:

$6x^2$ has $x^2$
$6x$ has $x$
15 has no variable.

Since the term 15 has no variable, there are no variables common to all three terms. Therefore, the variable part of the GCMF is just 1 (or we can say there's no variable part).

Step 4: Determine the Greatest Common Monomial Factor (GCMF)

Combining the GCF of the coefficients (3) and the common variables (none in this case), we find that the GCMF is 3.

Step 5: Factor out the GCMF

To factor out the GCMF, we divide each term of the polynomial by the GCMF and write the result in parentheses. The GCMF goes outside the parentheses.

$6x^2 + 6x + 15 = 3( rac{6x^2}{3} + rac{6x}{3} + rac{15}{3})$

Step 6: Simplify the Expression

Now, we simplify the fractions inside the parentheses:

$3( rac{6x^2}{3} + rac{6x}{3} + rac{15}{3}) = 3(2x^2 + 2x + 5)$

Step 7: Rewrite the Factored Expression

Therefore, the factored form of the polynomial $6x^2 + 6x + 15$ is $3(2x^2 + 2x + 5)$ .

Factoring out the Negative of the GCMF

Sometimes, it's beneficial to factor out the negative of the GCMF. This is often done when the leading coefficient (the coefficient of the term with the highest exponent) is negative. Factoring out the negative GCMF changes the signs of all the terms inside the parentheses.

Let's consider a slightly modified example: $-6x^2 - 6x - 15$ .

Following the same steps as before, we find that the GCF of the coefficients (6, 6, and 15) is 3. However, since the leading coefficient is negative, we'll factor out -3 (the negative of the GCMF).

$-6x^2 - 6x - 15 = -3( rac{-6x^2}{-3} + rac{-6x}{-3} + rac{-15}{-3})$

Simplifying, we get:

$-3(2x^2 + 2x + 5)$

Notice that all the signs inside the parentheses have changed. Factoring out the negative GCMF can be a useful technique in certain situations, especially when dealing with further factoring or solving equations.

Practice and Mastery

Factoring polynomials using the GCMF is a foundational skill that requires practice. The more you practice, the more comfortable you'll become with identifying the GCMF and factoring expressions efficiently. Here are some additional tips for mastering this technique:

Always look for the GCMF first: Before attempting any other factoring methods, always check if there's a GCMF that can be factored out. This simplifies the polynomial and makes further factoring easier.
Pay attention to signs: When factoring out the GCMF, be mindful of the signs of the terms. If the leading coefficient is negative, consider factoring out the negative GCMF.
Double-check your work: After factoring, you can always check your answer by distributing the GCMF back into the parentheses. If you get the original polynomial, your factoring is correct.
Practice with a variety of examples: Work through a range of examples with different coefficients, variables, and exponents. This will help you develop a strong understanding of the GCMF concept.

By understanding the concept of the greatest common monomial factor and practicing regularly, you'll develop a solid foundation for more advanced factoring techniques and algebraic concepts. Factoring is a crucial skill in mathematics, and mastering it will significantly enhance your problem-solving abilities.

In summary, factoring the polynomial $6x + 15 + 6x^2$ involves rearranging it to $6x^2 + 6x + 15$ , identifying the GCF of the coefficients as 3, recognizing the absence of common variables, factoring out the GCMF to get $3(2x^2 + 2x + 5)$ , and rewriting the expression in its factored form. This process demonstrates the fundamental steps in factoring polynomials using the greatest common monomial factor, a skill essential for algebraic manipulations and problem-solving.