Certainly! Let's tackle each part step-by-step to factor the given expression:
Given expression:
[tex]\[ 3x^3 y + 12x y - 9x^2 y - 36y \][/tex]
### Part A: Factoring out the Greatest Common Factor (GCF)
To begin, we need to identify the greatest common factor for all the terms in the expression.
Looking at each term:
- [tex]\( 3x^3 y \)[/tex]
- [tex]\( 12x y \)[/tex]
- [tex]\( 9x^2 y \)[/tex]
- [tex]\( 36y \)[/tex]
We notice that each term includes a factor of [tex]\( 3y \)[/tex]. Therefore, the greatest common factor (GCF) is [tex]\( 3y \)[/tex].
Next, we factor out [tex]\( 3y \)[/tex] from each term:
[tex]\[ 3y (x^3) + 3y (4x) - 3y (3x^2) - 3y (12) \][/tex]
When we factor [tex]\( 3y \)[/tex] out, we divide each term by [tex]\( 3y \)[/tex]:
[tex]\[
= 3y \left( \frac{3x^3 y}{3y} + \frac{12x y}{3y} - \frac{9x^2 y}{3y} - \frac{36y}{3y} \right)
\][/tex]
Simplifying the terms inside the parentheses:
[tex]\[
= 3y \left( x^3 + 4x - 3x^2 - 12 \right)
\][/tex]
Therefore, the expression with the greatest common factor factored out is:
[tex]\[
3y (x^3 - 3x^2 + 4x - 12)
\][/tex]
### Part B: Factoring the Entire Expression Completely
After factoring out the GCF in Part A, we now have:
[tex]\[
3y (x^3 - 3x^2 + 4x - 12)
\][/tex]
Our goal is to factor [tex]\( x^3 - 3x^2 + 4x - 12 \)[/tex] completely. We will use the method of grouping.
Step 1: Group the terms to facilitate factoring by grouping:
[tex]\[
x^3 - 3x^2 + 4x - 12 = (x^3 - 3x^2) + (4x - 12)
\][/tex]
Step 2: Factor out common factors from each group:
[tex]\[
= x^2 (x - 3) + 4 (x - 3)
\][/tex]
Step 3: Notice that [tex]\( (x - 3) \)[/tex] is a common factor in both groups. We factor [tex]\( (x - 3) \)[/tex] out:
[tex]\[
= (x - 3) (x^2 + 4)
\][/tex]
Now the entire expression is factored completely:
[tex]\[
= 3y (x - 3)(x^2 + 4)
\][/tex]
### Final Factored Form:
Thus, the completely factored form of the original expression is:
[tex]\[
3y (x - 3)(x^2 + 4)
\][/tex]
### Recap:
- Part A: Factored out the GCF [tex]\( 3y \)[/tex]
[tex]\[
3y (x^3 - 3x^2 + 4x - 12)
\][/tex]
- Part B: Completely factored the remaining polynomial by grouping and further factorization:
[tex]\[
3y (x - 3)(x^2 + 4)
\][/tex]
