Certainly! Let's proceed step-by-step to solve both parts of the problem.
### Part A: Factoring out the Greatest Common Factor (GCF)
Given expression:
[tex]\[ 3x^3y + 12xy - 9x^2y - 36y \][/tex]
1. Identify the GCF of each term in the expression.
- Terms:
- [tex]\( 3x^3y \)[/tex]
- [tex]\( 12xy \)[/tex]
- [tex]\( 9x^2y \)[/tex]
- [tex]\( 36y \)[/tex]
- Coefficient GCF: The coefficients are 3, 12, 9, and 36. The greatest common factor is 3.
- Variable GCF: Examining the variables, each term includes [tex]\( y \)[/tex] and the smallest power of [tex]\( x \)[/tex] present in any term is [tex]\( x^0 = 1 \)[/tex] (as [tex]\( 36y \)[/tex] does not contain [tex]\( x \)[/tex]). Hence, we take [tex]\( y \)[/tex] as the GCF for the variables.
2. Factor the GCF out of the expression.
- Extract the GCF, which is [tex]\( 3y \)[/tex], from each term:
[tex]\[ 3x^3y + 12xy - 9x^2y - 36y = 3y(x^3 + 4x - 3x^2 - 12) \][/tex]
So, the expression by factoring out the GCF is:
[tex]\[ 3y(x^3 + 4x - 3x^2 - 12) \][/tex]
### Part B: Factor the Entire Expression Completely
Now, we need to factor the expression inside the parentheses completely.
[tex]\[ x^3 + 4x - 3x^2 - 12 \][/tex]
1. Rearrange terms to group them in a way that makes factoring easier:
[tex]\[ x^3 - 3x^2 + 4x - 12 \][/tex]
2. Factor by grouping:
- Group the terms:
[tex]\[ (x^3 - 3x^2) + (4x - 12) \][/tex]
- Factor out the common factors from each group:
[tex]\[ x^2(x - 3) + 4(x - 3) \][/tex]
3. Factor out the common binomial [tex]\( (x - 3) \)[/tex]:
[tex]\[ (x^2 + 4)(x - 3) \][/tex]
So, the fully factored expression is:
[tex]\[ 3y (x^2 + 4)(x - 3) \][/tex]
### Summary
- Part A: The expression after factoring out the GCF is:
[tex]\[ 3y(x^3 + 4x - 3x^2 - 12) \][/tex]
- Part B: The completely factored expression is:
[tex]\[ 3y (x^2 + 4)(x - 3) \][/tex]
