Algebra 1 worksheet on factoring common monomial factors, designed for 1st Quarter assessment at Malabanan Integrated School.

Problem Overview:

The task involves factoring each of the given algebraic expressions by finding their Common Monomial Factor (CMF). The goal is to simplify each expression by extracting the greatest common factor (GCF) from all terms.

Solution:

#### EASY Section:

1. Expression: $ 8x + 16x^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(8, 16) = 8 $.
- Step 2: Identify the GCF of the variables: $ x $ is common in both terms.
- Step 3: Factor out $ 8x $:
$$
8x + 16x^2 = 8x(1 + 2x)
$$
- Answer: $ \boxed{8x(1 + 2x)} $

2. Expression: $ 6a^2 - 9a $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(6, 9) = 3 $.
- Step 2: Identify the GCF of the variables: $ a $ is common in both terms.
- Step 3: Factor out $ 3a $:
$$
6a^2 - 9a = 3a(2a - 3)
$$
- Answer: $ \boxed{3a(2a - 3)} $

3. Expression: $ 16p^4 + 4p^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(16, 4) = 4 $.
- Step 2: Identify the GCF of the variables: $ p^2 $ is common in both terms.
- Step 3: Factor out $ 4p^2 $:
$$
16p^4 + 4p^2 = 4p^2(4p^2 + 1)
$$
- Answer: $ \boxed{4p^2(4p^2 + 1)} $

4. Expression: $ 23m^3 + 5m $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(23, 5) = 1 $.
- Step 2: Identify the GCF of the variables: $ m $ is common in both terms.
- Step 3: Factor out $ m $:
$$
23m^3 + 5m = m(23m^2 + 5)
$$
- Answer: $ \boxed{m(23m^2 + 5)} $

5. Expression: $ 15n^3 + 12n $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(15, 12) = 3 $.
- Step 2: Identify the GCF of the variables: $ n $ is common in both terms.
- Step 3: Factor out $ 3n $:
$$
15n^3 + 12n = 3n(5n^2 + 4)
$$
- Answer: $ \boxed{3n(5n^2 + 4)} $

6. Expression: $ 20k^4 + 5k^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(20, 5) = 5 $.
- Step 2: Identify the GCF of the variables: $ k^2 $ is common in both terms.
- Step 3: Factor out $ 5k^2 $:
$$
20k^4 + 5k^2 = 5k^2(4k^2 + 1)
$$
- Answer: $ \boxed{5k^2(4k^2 + 1)} $

7. Expression: $ 13y^2 - 3y $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(13, 3) = 1 $.
- Step 2: Identify the GCF of the variables: $ y $ is common in both terms.
- Step 3: Factor out $ y $:
$$
13y^2 - 3y = y(13y - 3)
$$
- Answer: $ \boxed{y(13y - 3)} $

8. Expression: $ 6 + 10b $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(6, 10) = 2 $.
- Step 2: Identify the GCF of the variables: No variable is common.
- Step 3: Factor out $ 2 $:
$$
6 + 10b = 2(3 + 5b)
$$
- Answer: $ \boxed{2(3 + 5b)} $

9. Expression: $ 6d^2 + 10d^3 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(6, 10) = 2 $.
- Step 2: Identify the GCF of the variables: $ d^2 $ is common in both terms.
- Step 3: Factor out $ 2d^2 $:
$$
6d^2 + 10d^3 = 2d^2(3 + 5d)
$$
- Answer: $ \boxed{2d^2(3 + 5d)} $

10. Expression: $ 23mp + 10 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(23, 10) = 1 $.
- Step 2: Identify the GCF of the variables: No variable is common.
- Step 3: Since there is no common factor other than 1, the expression is already in its simplest form.
- Answer: $ \boxed{23mp + 10} $

11. Expression: $ 11k^2 + 8y $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(11, 8) = 1 $.
- Step 2: Identify the GCF of the variables: No variable is common.
- Step 3: Since there is no common factor other than 1, the expression is already in its simplest form.
- Answer: $ \boxed{11k^2 + 8y} $

#### AVERAGE Section:

12. Expression: $ 12x^2 + 6x - 36 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(12, 6, 36) = 6 $.
- Step 2: Identify the GCF of the variables: No variable is common across all terms.
- Step 3: Factor out $ 6 $:
$$
12x^2 + 6x - 36 = 6(2x^2 + x - 6)
$$
- Answer: $ \boxed{6(2x^2 + x - 6)} $

13. Expression: $ 14k^2y^2 + 2xy^2 - 6y^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(14, 2, 6) = 2 $.
- Step 2: Identify the GCF of the variables: $ y^2 $ is common in all terms.
- Step 3: Factor out $ 2y^2 $:
$$
14k^2y^2 + 2xy^2 - 6y^2 = 2y^2(7k^2 + x - 3)
$$
- Answer: $ \boxed{2y^2(7k^2 + x - 3)} $

14. Expression: $ 25a^3b^3 + 6a^2b^3 - 20ab^3 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(25, 6, 20) = 1 $.
- Step 2: Identify the GCF of the variables: $ ab^3 $ is common in all terms.
- Step 3: Factor out $ ab^3 $:
$$
25a^3b^3 + 6a^2b^3 - 20ab^3 = ab^3(25a^2 + 6a - 20)
$$
- Answer: $ \boxed{ab^3(25a^2 + 6a - 20)} $

15. Expression: $ 32m^2n - 48mn + 24 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(32, 48, 24) = 8 $.
- Step 2: Identify the GCF of the variables: $ mn $ is common in the first two terms, but not in the last term.
- Step 3: Factor out $ 8 $:
$$
32m^2n - 48mn + 24 = 8(4m^2n - 6mn + 3)
$$
- Answer: $ \boxed{8(4m^2n - 6mn + 3)} $

16. Expression: $ 24p^2 + 24pq^2 - 27q^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(24, 24, 27) = 3 $.
- Step 2: Identify the GCF of the variables: No variable is common across all terms.
- Step 3: Factor out $ 3 $:
$$
24p^2 + 24pq^2 - 27q^2 = 3(8p^2 + 8pq^2 - 9q^2)
$$
- Answer: $ \boxed{3(8p^2 + 8pq^2 - 9q^2)} $

17. Expression: $ 12z^2 + 15z^3 + 15y^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(12, 15, 15) = 3 $.
- Step 2: Identify the GCF of the variables: No variable is common across all terms.
- Step 3: Factor out $ 3 $:
$$
12z^2 + 15z^3 + 15y^2 = 3(4z^2 + 5z^3 + 5y^2)
$$
- Answer: $ \boxed{3(4z^2 + 5z^3 + 5y^2)} $

18. Expression: $ 21m^3n^2 + 27m^2n^2 + 12mn^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(21, 27, 12) = 3 $.
- Step 2: Identify the GCF of the variables: $ mn^2 $ is common in all terms.
- Step 3: Factor out $ 3mn^2 $:
$$
21m^3n^2 + 27m^2n^2 + 12mn^2 = 3mn^2(7m^2 + 9m + 4)
$$
- Answer: $ \boxed{3mn^2(7m^2 + 9m + 4)} $

#### HARD Section:

19. Expression: $ 24x^2y - 20xy^2 - 24x^3y + 28xy^3 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(24, 20, 24, 28) = 4 $.
- Step 2: Identify the GCF of the variables: $ xy $ is common in all terms.
- Step 3: Factor out $ 4xy $:
$$
24x^2y - 20xy^2 - 24x^3y + 28xy^3 = 4xy(6x - 5y - 6x^2 + 7y^2)
$$
- Answer: $ \boxed{4xy(6x - 5y - 6x^2 + 7y^2)} $

20. Expression: $ 20w^3u^2c - 18w^3u^3c^2 - 27w^3u^3c^3 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(20, 18, 27) = 1 $.
- Step 2: Identify the GCF of the variables: $ w^3u^2c $ is common in all terms.
- Step 3: Factor out $ w^3u^2c $:
$$
20w^3u^2c - 18w^3u^3c^2 - 27w^3u^3c^3 = w^3u^2c(20 - 18uc - 27uc^2)
$$
- Answer: $ \boxed{w^3u^2c(20 - 18uc - 27uc^2)} $

21. Expression: $ 22.5mm^2p + 5mm^3p^2 + 15mm^2p^3 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(22.5, 5, 15) = 2.5 $.
- Step 2: Identify the GCF of the variables: $ mm^2p $ is common in all terms.
- Step 3: Factor out $ 2.5mm^2p $:
$$
22.5mm^2p + 5mm^3p^2 + 15mm^2p^3 = 2.5mm^2p(9 + 2mp + 6p^2)
$$
- Answer: $ \boxed{2.5mm^2p(9 + 2mp + 6p^2)} $

22. Expression: $ 22.18dp + 66dp^2 + 72qp^2 - 24mp^2 $
- Step 1: Identify the GCF of the coefficients: $ \text{GCF}(22.18, 66, 72, 24) = 0.2 $.
- Step 2: Identify the GCF of the variables: $ p $ is common in all terms.
- Step 3: Factor out $ 0.2p $:
$$
22.18dp + 66dp^2 + 72qp^2 - 24mp^2 = 0.2p(110.9d + 330dp + 360qp - 120mp)
$$
- Answer: $ \boxed{0.2p(110.9d + 330dp + 360qp - 120mp)} $

Final Answer:

$$
\boxed{\text{See detailed solutions above.}}
$$