How is dividing by a monomial similar to factoring out the GCF?

Dividing by a monomial is similar to factoring out the greatest common factor (GCF) because both operations involve reducing a polynomial expression by a common factor.

To understand the similarity, let's break down each process:

1. Dividing by a monomial:
When dividing a polynomial expression by a monomial, you divide each term of the polynomial by the monomial. The monomial acts as a common factor that can be "cancelled out" from each term.

For example, consider the expression:
6x^3 - 12x^2 + 18x divided by 3x

Dividing each term by 3x, we get:
(6x^3/3x) - (12x^2/3x) + (18x/3x)

Simplifying further, we cancel out the common factor of 3x in each term, resulting in:
2x^2 - 4x + 6

In this case, dividing by the monomial 3x reduced each term by the common factor and simplified the expression.

2. Factoring out the GCF:
Factoring out the GCF involves identifying the highest common factor of all the terms in a polynomial expression and then dividing each term by that factor.

For example, consider the expression:
12x^3 - 18x^2 + 24x

The GCF of the terms is 6x, as it is the largest factor that can divide each term evenly. By factoring out the GCF, we divide each term by 6x:

6x(2x^2) - 6x(3x) + 6x(4)

After dividing, we are left with:
12x^2 - 18x + 24

As you can see, both operations result in the same simplified expression. Dividing by a monomial and factoring out the GCF both involve recognizing and reducing a common factor from each term of a polynomial expression to simplify it.