91Ó°ÊÓ

The Greatest Common Factor (GCF) is a key concept in simplifying and factoring expressions. To find the GCF, we look for the largest number that can divide each term in the expression without leaving a remainder. In this exercise, our terms are \(-18x\), \(-12y\), and \(6\). Let’s break this down:

- The prime factors of \(-18x\) are \(-1, 2, 3^2, x\)
- The prime factors of \(-12y\) are \(-1, 2^2, 3, y\)
- The prime factors of \(6\) are \(2, 3\)

To find the GCF, identify the common factors in all terms. Here, every term is divisible by \(6\). Therefore, the GCF of \(-18x, -12y,\) and \(6\) is \(6\). Recognizing and factoring out the GCF simplifies the expression, making further operations easier.

Factorization is the process of breaking down an expression into a product of simpler terms. After determining the GCF, we can factor out this common factor. Using the original expression \(-18x - 12y + 6\), we factor out the GCF \(6\):

\[-18x - 12y + 6 = 6(-3x - 2y + 1)\]

Each term inside the parentheses is what remains after dividing each term of the original expression by the GCF:

- Divide \(-18x\) by \(6\) to get \(-3x\)
- Divide \(-12y\) by \(6\) to get \(-2y\)
- Divide \(6\) by \(6\) to get \(1\)

Factoring expressions in this manner simplifies them and makes it easier to handle in further mathematical operations.

Multiplication verification ensures that the factored form is correct. It means checking our work by multiplying the factored form back out to see if we get the original expression. For our problem, we need to verify that

\[6(-3x - 2y + 1)\]

indeed equals \(-18x-12y+6\). We do this by distributing the \(6\) back into each term:

• Multiply \(6\) by \(-3x\) to get \(-18x\)


• Multiply \(6\) by \(-2y\) to get \(-12y\)


• Multiply \(6\) by \(1\) to get \(6\)

Combining these results, we see that

\[6(-3x - 2y + 1) = -18x - 12y + 6\]

Because the results match the original expression, our factorization is verified as correct. Always remember, verifying your factorization by multiplying ensures accuracy and reliability in your work.