Factoring Common Monomial worksheet with 14 algebraic expressions to factor completely.

Let's solve each problem by factoring out the greatest common monomial factor (GCF) from each expression. We'll identify the GCF of the coefficients and the lowest power of the variable(s) present in all terms, then factor it out.

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1. $ 6k^2 - 66k $

- GCF of 6 and 66 is 6
- Both terms have at least one $ k $
- So, GCF = $ 6k $
- Factor:
$$
6k(k - 11)
$$

✔ Answer: $ 6k(k - 11) $

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2. $ 6w^4 - 10w^3 + 2w $

- GCF of 6, 10, and 2 is 2
- All terms have at least one $ w $
- Lowest power of $ w $ is $ w^1 $
- So, GCF = $ 2w $
- Factor:
$$
2w(3w^3 - 5w^2 + 1)
$$

✔ Answer: $ 2w(3w^3 - 5w^2 + 1) $

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3. $ 11x^2 - 165x $

- GCF of 11 and 165 is 11
- Both terms have $ x $
- So, GCF = $ 11x $
- Factor:
$$
11x(x - 15)
$$

✔ Answer: $ 11x(x - 15) $

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4. $ 4p^2 - 32p $

- GCF of 4 and 32 is 4
- Both terms have $ p $
- So, GCF = $ 4p $
- Factor:
$$
4p(p - 8)
$$

✔ Answer: $ 4p(p - 8) $

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5. $ 2x^2 - 8x $

- GCF of 2 and 8 is 2
- Both terms have $ x $
- So, GCF = $ 2x $
- Factor:
$$
2x(x - 4)
$$

✔ Answer: $ 2x(x - 4) $

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6. $ 2y^5 - 18y^3 $

- GCF of 2 and 18 is 2
- Both terms have $ y^3 $ (lowest power)
- So, GCF = $ 2y^3 $
- Factor:
$$
2y^3(y^2 - 9)
$$

✔ Answer: $ 2y^3(y^2 - 9) $

> Note: $ y^2 - 9 $ is a difference of squares, but since the question asks only to factor completely, and we are factoring common monomials, this is sufficient unless further factoring is required. But since the instruction is "factor completely", we can go further:
>
> $ y^2 - 9 = (y - 3)(y + 3) $
>
> So fully factored:
> $$
> 2y^3(y - 3)(y + 3)
> $$

✔ Final Answer: $ 2y^3(y - 3)(y + 3) $

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7. $ 22m^2 - 99m $

- GCF of 22 and 99 is 11
- Both terms have $ m $
- So, GCF = $ 11m $
- Factor:
$$
11m(2m - 9)
$$

✔ Answer: $ 11m(2m - 9) $

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8. $ 4x^2 - 196x $

- GCF of 4 and 196 is 4
- Both terms have $ x $
- So, GCF = $ 4x $
- Factor:
$$
4x(x - 49)
$$

✔ Answer: $ 4x(x - 49) $

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9. $ x^4 - x^3 $

- GCF of both terms: $ x^3 $
- Factor:
$$
x^3(x - 1)
$$

✔ Answer: $ x^3(x - 1) $

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10. $ 2v^2 - 80v $

- GCF of 2 and 80 is 2
- Both terms have $ v $
- So, GCF = $ 2v $
- Factor:
$$
2v(v - 40)
$$

✔ Answer: $ 2v(v - 40) $

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11. $ 25m^2 + 5m $

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

✔ Answer: $ 5m(5m + 1) $

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12. $ 15y^5 - 3y^2 $

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

✔ Answer: $ 3y^2(5y^3 - 1) $

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13. $ 30p + 60 $

- GCF of 30 and 60 is 30
- No variable in both terms (only $ p $ in first), so GCF = 30
- Factor:
$$
30(p + 2)
$$

✔ Answer: $ 30(p + 2) $

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14. $ 4x^2 + 8x $

- GCF of 4 and 8 is 4
- Both terms have $ x $
- So, GCF = $ 4x $
- Factor:
$$
4x(x + 2)
$$

✔ Answer: $ 4x(x + 2) $

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## ✔ Final Answers:

| Problem | Factored Form |
|--------|----------------|
| 1 | $ 6k(k - 11) $ |
| 2 | $ 2w(3w^3 - 5w^2 + 1) $ |
| 3 | $ 11x(x - 15) $ |
| 4 | $ 4p(p - 8) $ |
| 5 | $ 2x(x - 4) $ |
| 6 | $ 2y^3(y - 3)(y + 3) $ |
| 7 | $ 11m(2m - 9) $ |
| 8 | $ 4x(x - 49) $ |
| 9 | $ x^3(x - 1) $ |
| 10 | $ 2v(v - 40) $ |
| 11 | $ 5m(5m + 1) $ |
| 12 | $ 3y^2(5y^3 - 1) $ |
| 13 | $ 30(p + 2) $ |
| 14 | $ 4x(x + 2) $ |

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