Certainly! Let's work through this step-by-step to factor out the greatest common factor (GCF) from the expression:
[tex]\[
6a^2xm^2 - 18a^5x^3m
\][/tex]
### Step 1: Identify the Greatest Common Factor (GCF)
First, we need to look at each term and determine the common factors.
#### For the coefficients in the terms:
- The numbers are [tex]\(6\)[/tex] and [tex]\(-18\)[/tex].
- The GCF of [tex]\(6\)[/tex] and [tex]\(-18\)[/tex] is [tex]\(6\)[/tex].
#### For the variables in the terms:
- In the first term, [tex]\(6a^2xm^2\)[/tex]:
[tex]\[
a^2 \quad \text{(meaning: } a \text{ to the power of } 2 \text{)}
\][/tex]
[tex]\[
x \quad \text{(meaning: } x \text{ to the power of } 1 \text{)}
\][/tex]
[tex]\[
m^2 \quad \text{(meaning: } m \text{ to the power of } 2 \text{)}
\][/tex]
- In the second term, [tex]\(-18a^5x^3m\)[/tex]:
[tex]\[
a^5 \quad \text{(meaning: } a \text{ to the power of } 5 \text{)}
\][/tex]
[tex]\[
x^3 \quad \text{(meaning: } x \text{ to the power of } 3 \text{)}
\][/tex]
[tex]\[
m \quad \text{(meaning: } m \text{ to the power of } 1 \text{)}
\][/tex]
#### Determine the lowest powers of common variables:
- For [tex]\(a\)[/tex], the lowest exponent is [tex]\(a^2\)[/tex].
- For [tex]\(x\)[/tex], the lowest exponent is [tex]\(x\)[/tex].
- For [tex]\(m\)[/tex], the lowest exponent is [tex]\(m\)[/tex].
So, the GCF of the variables is [tex]\(a^2xm\)[/tex].
### Step 2: Combine the GCF of the coefficients and variables
The overall GCF is:
[tex]\[
6a^2xm
\][/tex]
### Step 3: Factor out the GCF from each term
Now, we divide each term by [tex]\(6a^2xm\)[/tex] and factor it out.
[tex]\[
\begin{align*}
6a^2xm^2 & = 6a^2xm \cdot m, \\
-18a^5x^3m & = 6a^2xm \cdot (-3a^3x^2).
\end{align*}
\][/tex]
### Step 4: Write the factored expression
By factoring out [tex]\(6a^2xm\)[/tex], the expression becomes:
[tex]\[
6a^2xm (m - 3a^3x^2)
\][/tex]
Thus, the final factored form of the expression is:
[tex]\[
-6a^2xm (3a^3x^2 - m)
\][/tex]
Here, we see that the negative sign is within the grouping to conform with the conventional approach of leaving a positive leading coefficient in factored form.
So, the completely factored form of the given expression is:
[tex]\[
6a^2xm(m - 3a^3x^2)
\][/tex]
And the greatest common factor is:
[tex]\[
6a^2xm
\][/tex]
