Factoring the Common Monomial Factor Worksheet | PDF - Free Printable

Let’s solve each problem step by step. We’re factoring out the Common Monomial Factor (CMF) — that means we look for the biggest number and variable(s) that divide evenly into every term in the expression.

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EASY

1. 8a + 10a²
→ Coefficients: GCF of 8 and 10 is 2
→ Variables: both have at least one ‘a’ → a
→ CMF = 2a
→ Factor it out: 2a(4 + 5a)

2. 6n² – 9n
→ Coefficients: GCF of 6 and 9 is 3
→ Variables: both have at least one ‘n’ → n
→ CMF = 3n
→ Factor it out: 3n(2n – 3)

3. 16p⁴ + 4p⁶
→ Coefficients: GCF of 16 and 4 is 4
→ Variables: lowest power of p is p⁴ → p⁴
→ CMF = 4p⁴
→ Factor it out: 4p⁴(4 + p²)

4. 25m³ + 5m
→ Coefficients: GCF of 25 and 5 is 5
→ Variables: both have at least one ‘m’ → m
→ CMF = 5m
→ Factor it out: 5m(5m² + 1)

5. 15n³ + 12n
→ Coefficients: GCF of 15 and 12 is 3
→ Variables: both have at least one ‘n’ → n
→ CMF = 3n
→ Factor it out: 3n(5n² + 4)

6. 20x⁴ + 5x⁵
→ Coefficients: GCF of 20 and 5 is 5
→ Variables: lowest power of x is x⁴ → x⁴
→ CMF = 5x⁴
→ Factor it out: 5x⁴(4 + x)

7. 15y³ – 3y²
→ Coefficients: GCF of 15 and 3 is 3
→ Variables: lowest power of y is y² → y²
→ CMF = 3y²
→ Factor it out: 3y²(5y – 1)

8. 6 + 10b
→ Coefficients: GCF of 6 and 10 is 2
→ No common variable
→ CMF = 2
→ Factor it out: 2(3 + 5b)

9. 6h⁴ + 10h³
→ Coefficients: GCF of 6 and 10 is 2
→ Variables: lowest power of h is h³ → h³
→ CMF = 2h³
→ Factor it out: 2h³(3h + 5)

10. 25p + 10
→ Coefficients: GCF of 25 and 10 is 5
→ No common variable
→ CMF = 5
→ Factor it out: 5(5p + 2)

11. 4y³ + 8y
→ Coefficients: GCF of 4 and 8 is 4
→ Variables: both have at least one ‘y’ → y
→ CMF = 4y
→ Factor it out: 4y(y² + 2)

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AVERAGE

12. 32a⁴ + 8ab – 80b³
→ Coefficients: GCF of 32, 8, 80 is 8
→ Variables: no single variable appears in all terms → no variable CMF
→ CMF = 8
→ Factor it out: 8(4a⁴ + ab – 10b³)

13. 6x⁴y + 2xy³ – 6y
→ Coefficients: GCF of 6, 2, 6 is 2
→ Variables: all have at least one ‘y’ → y
→ CMF = 2y
→ Factor it out: 2y(3x⁴ + xy² – 3)

14. 20a¹²b⁴ + 6a⁹b – 20a⁷b⁵
→ Coefficients: GCF of 20, 6, 20 is 2
→ Variables:
- a: lowest power is a⁷ → a⁷
- b: lowest power is b → b
→ CMF = 2a⁷b
→ Factor it out: 2a⁷b(10a⁵b³ + 3 – 10b⁴)

15. 32mn³ – 48mn + 24
→ Coefficients: GCF of 32, 48, 24 is 8
→ Variables: not all terms have m or n → no variable CMF
→ CMF = 8
→ Factor it out: 8(4mn³ – 6mn + 3)

16. 24p³ + 24p²q – 27q²
→ Coefficients: GCF of 24, 24, 27 is 3
→ Variables: no single variable in all terms → no variable CMF
→ CMF = 3
→ Factor it out: 3(8p³ + 8p²q – 9q²)

17. 12x⁴ – 15x³y + 15y⁴
→ Coefficients: GCF of 12, 15, 15 is 3
→ Variables: no single variable in all terms → no variable CMF
→ CMF = 3
→ Factor it out: 3(4x⁴ – 5x³y + 5y⁴)

18. 21m⁴n³ + 27m³n² + 12m³n
→ Coefficients: GCF of 21, 27, 12 is 3
→ Variables:
- m: lowest power is m³ → m³
- n: lowest power is n → n
→ CMF = 3m³n
→ Factor it out: 3m³n(7mn² + 9n + 4)

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HARD

19. 4x⁴y – 20x³yz – 24x³y³ + 28x³yz
Wait — let’s check: last term is +28x³yz? That seems like a typo? But assuming it's correct:

Actually, looking again:
Terms:
- 4x⁴y
- –20x³yz
- –24x³y³
- +28x³yz ← this can combine with –20x³yz? But we’re factoring, not simplifying. Let’s factor as is.

→ Coefficients: GCF of 4, 20, 24, 28 is 4
→ Variables:
- x: lowest power is x³ → x³
- y: all have at least one y → y
- z: only in some terms → skip
→ CMF = 4x³y
→ Factor it out:
4x³y [ x – 5z – 6y² + 7z ] → simplify inside: x – 6y² + 2z
So: 4x³y(x – 6y² + 2z)

*Wait — let me double-check the original expression:*
“4x⁴y – 20x³yz – 24x³y³ + 28x³yz”

Combine like terms first? The problem says “find the factors”, so maybe we should combine before factoring?

–20x³yz + 28x³yz = +8x³yz

So expression becomes:
4x⁴y – 24x³y³ + 8x³yz

Now factor:

→ Coefficients: GCF of 4, 24, 8 is 4
→ Variables:
- x: lowest power x³ → x³
- y: all have at least one y → y
- z: only in last term → skip
→ CMF = 4x³y
→ Factor: 4x³y(x – 6y² + 2z)

✔ Final answer: 4x³y(x – 6y² + 2z)

20. 90a⁴b⁴c – 18a⁶b⁴ + 9a⁷b⁴c – 27a⁶b⁵
→ Coefficients: GCF of 90, 18, 9, 27 is 9
→ Variables:
- a: lowest power is a⁴ → a⁴
- b: lowest power is b⁴ → b⁴
- c: not in all terms → skip
→ CMF = 9a⁴b⁴
→ Factor it out:
9a⁴b⁴(10c – 2a² + a³c – 3ab)

Rearranged inside: 10c + a³c – 2a² – 3ab → but order doesn’t matter.

✔ Final answer: 9a⁴b⁴(10c – 2a² + a³c – 3ab)

21. 55m⁴np + 5m³np + 15mnp² + 25mnp
→ Coefficients: GCF of 55, 5, 15, 25 is 5
→ Variables:
- m: lowest power m → m
- n: all have n → n
- p: all have p → p
→ CMF = 5mnp
→ Factor it out:
5mnp(11m³ + m² + 3p + 5)

✔ Final answer: 5mnp(11m³ + m² + 3p + 5)

22. 18p + 60mp + 72pq² – 24mp²q²
→ Coefficients: GCF of 18, 60, 72, 24 is 6
→ Variables:
- p: all have at least one p → p
- m: not in first term → skip
- q: not in first two terms → skip
→ CMF = 6p
→ Factor it out:
6p(3 + 10m + 12q² – 4mpq²)

✔ Final answer: 6p(3 + 10m + 12q² – 4mpq²)

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Final Answer:

EASY:
1. 2a(4 + 5a)
2. 3n(2n – 3)
3. 4p⁴(4 + p²)
4. 5m(5m² + 1)
5. 3n(5n² + 4)
6. 5x⁴(4 + x)
7. 3y²(5y – 1)
8. 2(3 + 5b)
9. 2h³(3h + 5)
10. 5(5p + 2)
11. 4y(y² + 2)

AVERAGE:
12. 8(4a⁴ + ab – 10b³)
13. 2y(3x⁴ + xy² – 3)
14. 2a⁷b(10a⁵b³ + 3 – 10b⁴)
15. 8(4mn³ – 6mn + 3)
16. 3(8p³ + 8p²q – 9q²)
17. 3(4x⁴ – 5x³y + 5y⁴)
18. 3m³n(7mn² + 9n + 4)

HARD:
19. 4x³y(x – 6y² + 2z)
20. 9a⁴b⁴(10c – 2a² + a³c – 3ab)
21. 5mnp(11m³ + m² + 3p + 5)
22. 6p(3 + 10m + 12q² – 4mpq²)