Factoring the Common Monomial Factor Worksheet | PDF - Free Printable

Since I can't view or access images directly, I can't see the specific content of your uploaded image. However, based on the text you've provided, it appears to be a mathematics worksheet focused on factoring common monomial factors (CMF) in algebraic expressions.

Let me help you solve and explain each problem step-by-step. The worksheet has two sections: EASY and HARD, with 22 problems total.

---

🔷 What is Factoring Common Monomial Factor (CMF)?

Factoring out the common monomial factor means identifying the greatest common factor (GCF) of all terms in an expression and factoring it out. For example:

> $ 6x^2 + 3x = 3x(2x + 1) $

Here, $ 3x $ is the GCF of both terms.

We’ll go through each problem.

---

## ✔ EASY SECTION

1. $ 8a + 10b^2 $

- GCF of 8 and 10 is 2
- No common variable
- So, factor out 2:
$ = 2(4a + 5b^2) $

✔ Answer: $ 2(4a + 5b^2) $

---

2. $ 6n^2 - 9n $

- GCF of 6 and 9 is 3
- Both terms have $ n $, so GCF is $ 3n $
- Factor out $ 3n $:
$ = 3n(2n - 3) $

✔ Answer: $ 3n(2n - 3) $

---

3. $ 16p^2 + 6p^3 $

- GCF of 16 and 6 is 2
- Both terms have $ p^2 $, so GCF is $ 2p^2 $
- Factor out $ 2p^2 $:
$ = 2p^2(8 + 3p) $

✔ Answer: $ 2p^2(8 + 3p) $

---

4. $ 25m^2 + 5m $

- GCF of 25 and 5 is 5
- Both have $ m $, so GCF is $ 5m $
- $ = 5m(5m + 1) $

✔ Answer: $ 5m(5m + 1) $

---

5. $ 15n^3 + 12n $

- GCF of 15 and 12 is 3
- Both have $ n $, so GCF is $ 3n $
- $ = 3n(5n^2 + 4) $

✔ Answer: $ 3n(5n^2 + 4) $

---

6. $ 20x^2 + 5x^3 $

- GCF of 20 and 5 is 5
- Both have $ x^2 $, so GCF is $ 5x^2 $
- $ = 5x^2(4 + x) $

✔ Answer: $ 5x^2(4 + x) $

---

7. $ 15y^3 - 3y^2 $

- GCF of 15 and 3 is 3
- Both have $ y^2 $, so GCF is $ 3y^2 $
- $ = 3y^2(5y - 1) $

✔ Answer: $ 3y^2(5y - 1) $

---

8. $ 6 + 10b $

- GCF of 6 and 10 is 2
- No common variable
- $ = 2(3 + 5b) $

✔ Answer: $ 2(3 + 5b) $

---

9. $ 6h^2 + 10h^3 $

- GCF of 6 and 10 is 2
- Both have $ h^2 $, so GCF is $ 2h^2 $
- $ = 2h^2(3 + 5h) $

✔ Answer: $ 2h^2(3 + 5h) $

---

10. $ 25p + 10 $

- GCF of 25 and 10 is 5
- No common variable
- $ = 5(5p + 2) $

✔ Answer: $ 5(5p + 2) $

---

11. $ 6y^2 + 8y $

- GCF of 6 and 8 is 2
- Both have $ y $, so GCF is $ 2y $
- $ = 2y(3y + 4) $

✔ Answer: $ 2y(3y + 4) $

---

## ✔ AVERAGE SECTION

12. $ 32a^2 + 8ab - 80b^2 $

- GCF of 32, 8, 80 is 8
- No common variable across all terms
- $ = 8(4a^2 + ab - 10b^2) $

✔ Answer: $ 8(4a^2 + ab - 10b^2) $

---

13. $ 6x^3y + 2xy^2 - 6y $

- All terms have $ y $, so factor out $ y $
- Coefficients: GCF of 6, 2, 6 is 2
- So GCF is $ 2y $
- $ = 2y(3x^3 + x y - 3) $

Wait: Let's check:
- $ 6x^3y ÷ 2y = 3x^3 $
- $ 2xy^2 ÷ 2y = x y $
- $ -6y ÷ 2y = -3 $
- So yes: $ = 2y(3x^3 + xy - 3) $

✔ Answer: $ 2y(3x^3 + xy - 3) $

---

14. $ 20a^3b^2 + 6a^2b^3 - 20a^2b^3 $

- First, combine like terms: $ 6a^2b^3 - 20a^2b^3 = -14a^2b^3 $
- So expression becomes: $ 20a^3b^2 - 14a^2b^3 $
- GCF of 20 and 14 is 2
- Both terms have $ a^2b^2 $
- GCF = $ 2a^2b^2 $
- $ = 2a^2b^2(10a - 7b) $

✔ Answer: $ 2a^2b^2(10a - 7b) $

---

15. $ 32mn^2 - 40mn + 24 $

- GCF of 32, 40, 24 is 8
- Only $ mn $ is common in first two terms, not third
- So GCF is 8
- $ = 8(4mn^2 - 5mn + 3) $

✔ Answer: $ 8(4mn^2 - 5mn + 3) $

---

16. $ 24p^4 + 24p^3q - 27q^2 $

- GCF of 24, 24, 27 is 3
- No common variable in all terms
- So GCF = 3
- $ = 3(8p^4 + 8p^3q - 9q^2) $

✔ Answer: $ 3(8p^4 + 8p^3q - 9q^2) $

---

17. $ 12a^2 - 15a^2y^4 + 15y^3 $

- GCF of 12, 15, 15 is 3
- Terms: $ 12a^2 $, $ -15a^2y^4 $, $ 15y^3 $
- No common variable in all three
- So GCF = 3
- $ = 3(4a^2 - 5a^2y^4 + 5y^3) $

✔ Answer: $ 3(4a^2 - 5a^2y^4 + 5y^3) $

---

18. $ 21m^2n^4 + 27m^2n^2 + 12m^3n $

- GCF of coefficients: 21, 27, 12 → GCF is 3
- Variables: all have $ m^2 $, but $ n $ powers vary
- Lowest power of $ n $ is 1? Wait:
- $ m^2n^4 $, $ m^2n^2 $, $ m^3n $ → common is $ m^2n $
- So GCF = $ 3m^2n $
- Now divide:
- $ 21m^2n^4 ÷ 3m^2n = 7n^3 $
- $ 27m^2n^2 ÷ 3m^2n = 9n $
- $ 12m^3n ÷ 3m^2n = 4m $
- So: $ = 3m^2n(7n^3 + 9n + 4m) $

✔ Answer: $ 3m^2n(7n^3 + 9n + 4m) $

---

## ✔ HARD SECTION

19. $ 4x^3y - 20x^2yz - 24x^3y^2 + 20x^2yz $

- First, simplify: $ -20x^2yz + 20x^2yz = 0 $
- So only remaining: $ 4x^3y - 24x^3y^2 $
- GCF of 4 and 24 is 4
- Both have $ x^3y $
- So GCF = $ 4x^3y $
- $ = 4x^3y(1 - 6y) $

✔ Answer: $ 4x^3y(1 - 6y) $

---

20. $ 90a^3b^3c - 18a^2b^3 + 9a^2b^2c - 27a^2b^3 $

- Combine like terms:
- $ -18a^2b^3 - 27a^2b^3 = -45a^2b^3 $
- So expression becomes:
$ 90a^3b^3c - 45a^2b^3 + 9a^2b^2c $
- GCF of 90, 45, 9 is 9
- Variables: all have $ a^2b^2 $
- So GCF = $ 9a^2b^2 $
- Now divide:
- $ 90a^3b^3c ÷ 9a^2b^2 = 10ab c $
- $ -45a^2b^3 ÷ 9a^2b^2 = -5b $
- $ 9a^2b^2c ÷ 9a^2b^2 = c $
- So: $ = 9a^2b^2(10abc - 5b + c) $

✔ Answer: $ 9a^2b^2(10abc - 5b + c) $

---

21. $ 55mn^3p + 5m^2np + 15mnp^2 + 25mnp $

- GCF of coefficients: 55, 5, 15, 25 → GCF is 5
- Variables: all have $ m $, $ n $, $ p $
- Lowest powers: $ m^1 $, $ n^1 $, $ p^1 $
- So GCF = $ 5mnp $
- Divide:
- $ 55mn^3p ÷ 5mnp = 11n^2 $
- $ 5m^2np ÷ 5mnp = m $
- $ 15mnp^2 ÷ 5mnp = 3p $
- $ 25mnp ÷ 5mnp = 5 $
- So: $ = 5mnp(11n^2 + m + 3p + 5) $

✔ Answer: $ 5mnp(11n^2 + m + 3p + 5) $

---

22. $ 18p + 60mp + 72pq^3 - 24mp^2q^3 $

- GCF of 18, 60, 72, 24 → GCF is 6
- Variables: only $ p $ is common in all terms
- So GCF = $ 6p $
- Divide:
- $ 18p ÷ 6p = 3 $
- $ 60mp ÷ 6p = 10m $
- $ 72pq^3 ÷ 6p = 12q^3 $
- $ -24mp^2q^3 ÷ 6p = -4mpq^3 $
- So: $ = 6p(3 + 10m + 12q^3 - 4mpq^3) $

✔ Answer: $ 6p(3 + 10m + 12q^3 - 4mpq^3) $

---

## ✔ Summary of Key Steps

To factor CMF:
1. Find the greatest common numerical factor.
2. Identify the lowest power of each variable present in all terms.
3. Factor that GCF out of every term.
4. Write the expression as: $ \text{GCF} \times (\text{remaining terms}) $

---

If you need this in a printable format or want to verify any specific answer, let me know!