Let's solve each expression by factoring them step-by-step.
### Expression I: [tex]\( 18x^4y + 9xy^7 \)[/tex]
1. Identify the Greatest Common Factor (GCF):
Both terms share the common factor [tex]\( 9xy \)[/tex]:
- [tex]\( 18x^4y = 9xy \cdot 2x^3 \)[/tex]
- [tex]\( 9xy^7 = 9xy \cdot y^6 \)[/tex]
2. Factor out the GCF:
[tex]\[
18x^4y + 9xy^7 = 9xy(2x^3 + y^6)
\][/tex]
### Expression L: [tex]\( 6x^4 - 10x^3 + 2x \)[/tex]
1. Identify the GCF:
All terms share the common factor [tex]\( x \)[/tex]:
- [tex]\( 6x^4 = x \cdot 6x^3 \)[/tex]
- [tex]\( -10x^3 = x \cdot (-10x^2) \)[/tex]
- [tex]\( 2x = x \cdot 2 \)[/tex]
2. Factor out the GCF:
[tex]\[
6x^4 - 10x^3 + 2x = x(6x^3 - 10x^2 + 2)
\][/tex]
### Expression N: [tex]\( 8y^5 - 24y^4 - 16y^2 \)[/tex]
1. Identify the GCF:
All terms share the common factor [tex]\( 8y^2 \)[/tex]:
- [tex]\( 8y^5 = 8y^2 \cdot y^3 \)[/tex]
- [tex]\( -24y^4 = 8y^2 \cdot (-3y^2) \)[/tex]
- [tex]\( -16y^2 = 8y^2 \cdot (-2) \)[/tex]
2. Factor out the GCF:
[tex]\[
8y^5 - 24y^4 - 16y^2 = 8y^2(y^3 - 3y^2 - 2)
\][/tex]
### Fourth Expression: [tex]\( 5x^3y - 15x^2y^2 + 25x^2y \)[/tex]
1. Identify the GCF:
All terms share the common factor [tex]\( 5x^2y \)[/tex]:
- [tex]\( 5x^3y = 5x^2y \cdot x \)[/tex]
- [tex]\( -15x^2y^2 = 5x^2y \cdot (-3y) \)[/tex]
- [tex]\( 25x^2y = 5x^2y \cdot 5 \)[/tex]
2. Factor out the GCF:
[tex]\[
5x^3y - 15x^2y^2 + 25x^2y = 5x^2y(x - 3y + 5)
\][/tex]
### Summary of Factorized Forms:
1. For [tex]\( 18x^4y + 9xy^7 \)[/tex]:
[tex]\[
18x^4y + 9xy^7 = 9xy(2x^3 + y^6)
\][/tex]
2. For [tex]\( 6x^4 - 10x^3 + 2x \)[/tex]:
[tex]\[
6x^4 - 10x^3 + 2x = x(6x^3 - 10x^2 + 2)
\][/tex]
3. For [tex]\( 8y^5 - 24y^4 - 16y^2 \)[/tex]:
[tex]\[
8y^5 - 24y^4 - 16y^2 = 8y^2(y^3 - 3y^2 - 2)
\][/tex]
4. For [tex]\( 5x^3y - 15x^2y^2 + 25x^2y \)[/tex]:
[tex]\[
5x^3y - 15x^2y^2 + 25x^2y = 5x^2y(x - 3y + 5)
\][/tex]
