Question

1. $6a^{2}+27$
2. $4y^{2}-24y$

$4y(y-6)$

1. $21cd-3d$

$3d(7c-1)$

1. $14gh-18h$

$2h(7g-9$

1. $15a^{2}b-30ab$
2. $16bc^{2}+24bc$
3. $ab-a$
4. $x^{2}y+3xy$
5. $5x-13y$
6. $18a^{2}bc^{2}-48abc^{3}$
7. $2x^{2}y-2xy^{2}+4xy$
8. $9r^{8}-18r^{2}s-24rs^{2}$
9. $6y^{4}+14y^{3}-10y^{2}$
10. $12a^{5}b^{2}-36a^{4}b^{3}-6a^{2}b^{2}$
11. $14gh^{2}+28gh+14h$
12. $18x^{2}yz-24xz^{2}+36yz^{3}$
13. $m^{3}n-m^{2}n^{2}+5mn^{3}$
14. $16xy^{2}+28xy+8y$
15. $35a^{2}-20ab^{2}+15a$
16. $3a^{3}b^{2}c-9a^{2}b^{3}c^{2}+15ab^{4}c^{3}$

Explanation:

Step1: Factor out GCF of $6a^2+27$

$\gcd(6,27)=3$, so $3(2a^2 + 9)$

Step2: Verify Q6 completion

$4y^2-24y=4y(y-6)$ (correct as written)

Step3: Verify Q7 completion

$21cd-3d=3d(7c-1)$ (correct as written)

Step4: Fix Q8 missing parenthesis

$\gcd(14,18)=2$, common term $h$, so $2h(7g - 9)$

Step5: Factor Q9: $15a^2b-30ab$

$\gcd(15,30)=15$, common terms $ab$, so $15ab(a - 2)$

Step6: Factor Q10: $16bc^2+24bc$

$\gcd(16,24)=8$, common terms $bc$, so $8bc(2c + 3)$

Step7: Factor Q11: $ab-a$

Common term $a$, so $a(b - 1)$

Step8: Factor Q12: $x^2y+3xy$

Common terms $xy$, so $xy(x + 3)$

Step9: Note Q13 has no common factor

$5x-13y$ (already in simplest form)

Step10: Factor Q14: $18a^2bc^2-48abc^3$

$\gcd(18,48)=6$, common terms $abc^2$, so $6abc^2(3a - 8c)$

Step11: Factor Q15: $2x^2y-2xy^2+4xy$

$\gcd(2,2,4)=2$, common terms $xy$, so $2xy(x - y + 2)$

Step12: Factor Q16: $9r^8-18r^2s-24rs^2$

$\gcd(9,18,24)=3$, so $3(3r^8 - 6r^2s - 8rs^2)$

Step13: Factor Q17: $6y^4+14y^3-10y^2$

$\gcd(6,14,10)=2$, common term $y^2$, so $2y^2(3y^2 + 7y - 5)$

Step14: Factor Q18: $12a^5b^2-36a^4b^3-6a^2b^2$

$\gcd(12,36,6)=6$, common terms $a^2b^2$, so $6a^2b^2(2a^3 - 6a^2b - 1)$

Step15: Factor Q19: $14gh^2+28gh+14h$

$\gcd(14,28,14)=14$, common term $h$, so $14h(gh + 2g + 1)$

Step16: Factor Q20: $18x^2yz-24xz^2+36yz^3$

$\gcd(18,24,36)=6$, so $6(3x^2yz - 4xz^2 + 6yz^3)$

Step17: Factor Q21: $m^3n-m^2n^2+5mn^3$

Common terms $mn$, so $mn(m^2 - mn + 5n^2)$

Step18: Factor Q22: $16xy^2+28xy+8y$

$\gcd(16,28,8)=4$, common term $y$, so $4y(4xy + 7x + 2)$

Step19: Factor Q23: $35a^2-20ab^2+15a$

$\gcd(35,20,15)=5$, common term $a$, so $5a(7a - 4b^2 + 3)$

Step20: Factor Q24: $3a^3b^2c-9a^2b^3c^2+15ab^4c^3$

$\gcd(3,9,15)=3$, common terms $ab^2c$, so $3ab^2c(a^2 - 3abc + 5b^2c^2)$

Answer:

1. $3(2a^2 + 9)$
2. $4y(y-6)$
3. $3d(7c-1)$
4. $2h(7g - 9)$
5. $15ab(a - 2)$
6. $8bc(2c + 3)$
7. $a(b - 1)$
8. $xy(x + 3)$
9. $5x-13y$
10. $6abc^2(3a - 8c)$
11. $2xy(x - y + 2)$
12. $3(3r^8 - 6r^2s - 8rs^2)$
13. $2y^2(3y^2 + 7y - 5)$
14. $6a^2b^2(2a^3 - 6a^2b - 1)$
15. $14h(gh + 2g + 1)$
16. $6(3x^2yz - 4xz^2 + 6yz^3)$
17. $mn(m^2 - mn + 5n^2)$
18. $4y(4xy + 7x + 2)$
19. $5a(7a - 4b^2 + 3)$
20. $3ab^2c(a^2 - 3abc + 5b^2c^2)$