Algebra 1 worksheet on factoring common monomial factors, designed for 1st Quarter of the 2018-2019 school year.

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

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🔷 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.

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## ✔ 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) $

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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) $

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3. $ 16p^2 + 4p^3 $

- GCF of 16 and 4 is 4
- Both terms have $ p^2 $, so GCF is $ 4p^2 $
- Factor out $ 4p^2 $:
$ = 4p^2(4 + p) $

✔ Answer: $ 4p^2(4 + p) $

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4. $ 25m^2 + 5m $

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

✔ Answer: $ 5m(5m + 1) $

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5. $ 15n^3 + 12n $

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

✔ Answer: $ 3n(5n^2 + 4) $

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

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

✔ Answer: $ 5x^2(4 + x) $

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

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

✔ Answer: $ 3y^2(5y - 1) $

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8. $ 6 + 10b $

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

✔ Answer: $ 2(3 + 5b) $

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9. $ 6h^2 + 12h^3 $

- GCF of 6 and 12 is 6
- Both have $ h^2 $, so GCF is $ 6h^2 $
- Factor out $ 6h^2 $:
$ = 6h^2(1 + 2h) $

✔ Answer: $ 6h^2(1 + 2h) $

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10. $ 25p + 10 $

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

✔ Answer: $ 5(5p + 2) $

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11. $ 6y^2 + 8y $

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

✔ Answer: $ 2y(3y + 4) $

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## ✔ AVERAGE SECTION

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

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

✔ Answer: $ 8(4a^2 + ab - 10b^2) $

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

- All terms have $ y $
- Coefficients: 6, 2, 6 → GCF is 2
- So GCF is $ 2y $
- Factor out $ 2y $:
$ = 2y(3x^2 + xy - 3) $

✔ Answer: $ 2y(3x^2 + xy - 3) $

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14. $ 20a^3b^2 + 6a^2b^3 - 28a^2b^2 $

- GCF of coefficients: 20, 6, 28 → GCF is 2
- Variables:
- $ a^2 $ is common (min power)
- $ b^2 $ is common
- So GCF is $ 2a^2b^2 $
- Factor out $ 2a^2b^2 $:
$ = 2a^2b^2(10a + 3b - 14) $

✔ Answer: $ 2a^2b^2(10a + 3b - 14) $

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

- GCF of 32, 40, 24 is 8
- $ m $ is in first two terms but not third → no common variable for all
- So GCF is 8
- Factor out 8:
$ = 8(4mn^2 - 5mn + 3) $

✔ Answer: $ 8(4mn^2 - 5mn + 3) $

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

- GCF of 24, 24, 27 is 3
- $ p $ is not in last term → no common variable
- So GCF is 3
- Factor out 3:
$ = 3(8p^4 + 8p^3q - 9q^2) $

✔ Answer: $ 3(8p^4 + 8p^3q - 9q^2) $

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

- GCF of 12, 15, 15 is 3
- $ a^2 $ only in first two terms
- $ y^2 $ only in second and third
- No common variable across all → GCF is 3
- Factor out 3:
$ = 3(4a^2 - 5a^2y^2 + 5y^3) $

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

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

- GCF of 21, 27, 12 is 3
- $ m^2 $ is common (min power)
- $ n $ is common (min power is $ n $)
- So GCF is $ 3m^2n $
- Factor out $ 3m^2n $:
$ = 3m^2n(7mn + 9n + 4m) $

✔ Answer: $ 3m^2n(7mn + 9n + 4m) $

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## ✔ HARD SECTION

19. $ 4x^3y - 20x^2yz - 24x^3y^2 + 28x^2yz $

- Group terms:
$ = (4x^3y - 20x^2yz) + (-24x^3y^2 + 28x^2yz) $
- But better: look for overall GCF
- Coefficients: 4, 20, 24, 28 → GCF is 4
- $ x^2 $ is common in all (min power)
- $ y $ is in all? Yes → min power $ y $
- $ z $? Only in some → not common
- So GCF is $ 4x^2y $
- Now factor:
$ = 4x^2y(x - 5z - 6xy + 7z) $
Wait — simplify inside:
$ = 4x^2y(x - 6xy + 2z) $ → wait, let’s recheck:

Actually:
- $ 4x^3y ÷ 4x^2y = x $
- $ -20x^2yz ÷ 4x^2y = -5z $
- $ -24x^3y^2 ÷ 4x^2y = -6xy $
- $ 28x^2yz ÷ 4x^2y = 7z $

So:
$ = 4x^2y(x - 5z - 6xy + 7z) $

Now combine like terms: $ -5z + 7z = 2z $

So:
$ = 4x^2y(x - 6xy + 2z) $

But this is messy. Better to write as:
$ = 4x^2y(x - 6xy + 2z) $

Alternatively, rearrange:
$ = 4x^2y(-6xy + x + 2z) $

But standard form is fine.

✔ Answer: $ 4x^2y(x - 6xy + 2z) $

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

- Combine like terms:
- $ 90a^3b^2c + 9a^3b^2c = 99a^3b^2c $
- $ -18a^2b^3 - 27a^2b^3 = -45a^2b^3 $
- So expression becomes: $ 99a^3b^2c - 45a^2b^3 $
- GCF of 99 and 45 is 9
- $ a^2 $, $ b^2 $, common
- So GCF is $ 9a^2b^2 $
- Factor:
$ = 9a^2b^2(11ac - 5b) $

✔ Answer: $ 9a^2b^2(11ac - 5b) $

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21. $ 55mn^3p + 5m^2np + 15mnp^4 + 25mnp $

- GCF of coefficients: 55, 5, 15, 25 → GCF is 5
- $ m $ in all
- $ n $ in all
- $ p $ in all
- Min powers: $ m^1, n^1, p^1 $
- So GCF is $ 5mnp $
- Factor:
- $ 55mn^3p ÷ 5mnp = 11n^2 $
- $ 5m^2np ÷ 5mnp = m $
- $ 15mnp^4 ÷ 5mnp = 3p^3 $
- $ 25mnp ÷ 5mnp = 5 $
- So:
$ = 5mnp(11n^2 + m + 3p^3 + 5) $

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

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

- GCF of 18, 60, 72, 24 → GCF is 6
- $ p $ in all terms
- $ m $? Not in first or third → not common
- $ q $? Not in first or second → not common
- So GCF is 6p
- Factor out $ 6p $:
- $ 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) $

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## ✔ Final Answers Summary

| Problem | Answer |
|--------|--------|
| 1 | $ 2(4a + 5b^2) $ |
| 2 | $ 3n(2n - 3) $ |
| 3 | $ 4p^2(4 + p) $ |
| 4 | $ 5m(5m + 1) $ |
| 5 | $ 3n(5n^2 + 4) $ |
| 6 | $ 5x^2(4 + x) $ |
| 7 | $ 3y^2(5y - 1) $ |
| 8 | $ 2(3 + 5b) $ |
| 9 | $ 6h^2(1 + 2h) $ |
| 10 | $ 5(5p + 2) $ |
| 11 | $ 2y(3y + 4) $ |
| 12 | $ 8(4a^2 + ab - 10b^2) $ |
| 13 | $ 2y(3x^2 + xy - 3) $ |
| 14 | $ 2a^2b^2(10a + 3b - 14) $ |
| 15 | $ 8(4mn^2 - 5mn + 3) $ |
| 16 | $ 3(8p^4 + 8p^3q - 9q^2) $ |
| 17 | $ 3(4a^2 - 5a^2y^2 + 5y^3) $ |
| 18 | $ 3m^2n(7mn + 9n + 4m) $ |
| 19 | $ 4x^2y(x - 6xy + 2z) $ |
| 20 | $ 9a^2b^2(11ac - 5b) $ |
| 21 | $ 5mnp(11n^2 + m + 3p^3 + 5) $ |
| 22 | $ 6p(3 + 10m + 12q^3 - 4mpq^3) $ |

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📌 Tips for Factoring CMF:

1. Look at coefficients – find GCF of numbers.
2. Look at variables – take the lowest power that appears in all terms.
3. Factor out the GCF and write what's left inside parentheses.
4. Check by distributing back to verify.

Let me know if you'd like these answers formatted into a downloadable PDF or need explanations for any specific problem!