Factoring out the GCF worksheet with ten algebraic expressions to solve.

Let's solve each problem by factoring out the Greatest Common Factor (GCF) from each expression. We'll go step-by-step for each one.

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1. $ 15x^3 - 25x^2 + 55x $

- Coefficients: 15, 25, 55 → GCF is 5
- Variables: All terms have at least $ x $, so lowest power is $ x $
- GCF = $ 5x $

Factor:
$$
15x^3 - 25x^2 + 55x = 5x(3x^2 - 5x + 11)
$$

✔ Answer: $ 5x(3x^2 - 5x + 11) $

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2. $ 36x^3 + 63x^2 - 27x $

- Coefficients: 36, 63, 27 → GCF is 9
- All terms have $ x $ → lowest power is $ x $
- GCF = $ 9x $

Factor:
$$
36x^3 + 63x^2 - 27x = 9x(4x^2 + 7x - 3)
$$

✔ Answer: $ 9x(4x^2 + 7x - 3) $

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3. $ 25x^{12} - 35x^6 $

- Coefficients: 25, 35 → GCF is 5
- Variables: $ x^{12} $ and $ x^6 $ → lowest power is $ x^6 $
- GCF = $ 5x^6 $

Factor:
$$
25x^{12} - 35x^6 = 5x^6(5x^6 - 7)
$$

✔ Answer: $ 5x^6(5x^6 - 7) $

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4. $ -6p^5 - 6p^4 $

- Coefficients: -6, -6 → GCF is -6
- Variables: $ p^5 $, $ p^4 $ → lowest power is $ p^4 $
- GCF = $ -6p^4 $

Factor:
$$
-6p^5 - 6p^4 = -6p^4(p + 1)
$$

✔ Answer: $ -6p^4(p + 1) $

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5. $ 72x^5 - 72x^3 - 80x^2 $

- Coefficients: 72, 72, 80 → GCF of 72 and 80 is 8
- Variables: all have $ x^2 $ → lowest power is $ x^2 $
- GCF = $ 8x^2 $

Factor:
$$
72x^5 - 72x^3 - 80x^2 = 8x^2(9x^3 - 9x - 10)
$$

✔ Answer: $ 8x^2(9x^3 - 9x - 10) $

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6. $ 6x^2yz + 2xy^2z - 4xyz $

- Coefficients: 6, 2, 4 → GCF is 2
- Variables: Each term has $ x, y, z $ → lowest powers: $ x^1, y^1, z^1 $
- GCF = $ 2xyz $

Factor:
$$
6x^2yz + 2xy^2z - 4xyz = 2xyz(3x + y - 2)
$$

✔ Answer: $ 2xyz(3x + y - 2) $

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7. $ -16p^3q^2 + 24p^2q^3 - 32p^4q $

- Coefficients: 16, 24, 32 → GCF is 8
- Signs: all negative? No — first term is negative, others positive → factor out -8 to make leading term positive
- Variables: $ p^3, p^2, p^4 $ → lowest $ p^2 $; $ q^2, q^3, q $ → lowest $ q $
- GCF = $ -8p^2q $

Factor:
$$
-16p^3q^2 + 24p^2q^3 - 32p^4q = -8p^2q(2pq - 3q^2 + 4p^2)
$$

Note: Rearranged inside to match standard form (but not required).

✔ Answer: $ -8p^2q(2pq - 3q^2 + 4p^2) $

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8. $ 7wx(a - 9) - 10w(9 - a) $

Notice: $ (9 - a) = -(a - 9) $

So rewrite:
$$
7wx(a - 9) - 10w(9 - a) = 7wx(a - 9) + 10w(a - 9)
$$
Because $ -10w(9 - a) = +10w(a - 9) $

Now both terms have $ w(a - 9) $

Factor:
$$
= w(a - 9)(7x + 10)
$$

✔ Answer: $ w(a - 9)(7x + 10) $

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9. $ 36w^5 + 40w^4 - 160w^3 - 20w^2 $

- Coefficients: 36, 40, 160, 20 → GCF is 4
- Variables: all have $ w^2 $ → lowest power is $ w^2 $
- GCF = $ 4w^2 $

Factor:
$$
36w^5 + 40w^4 - 160w^3 - 20w^2 = 4w^2(9w^3 + 10w^2 - 40w - 5)
$$

✔ Answer: $ 4w^2(9w^3 + 10w^2 - 40w - 5) $

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10. $ bc(m + 10) - 5b(10 + m) $

Note: $ (10 + m) = (m + 10) $

So:
$$
bc(m + 10) - 5b(m + 10)
$$

Now both terms have $ b(m + 10) $

Factor:
$$
= b(m + 10)(c - 5)
$$

✔ Answer: $ b(m + 10)(c - 5) $

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✔ Final Answers:

1. $ 5x(3x^2 - 5x + 11) $
2. $ 9x(4x^2 + 7x - 3) $
3. $ 5x^6(5x^6 - 7) $
4. $ -6p^4(p + 1) $
5. $ 8x^2(9x^3 - 9x - 10) $
6. $ 2xyz(3x + y - 2) $
7. $ -8p^2q(2pq - 3q^2 + 4p^2) $
8. $ w(a - 9)(7x + 10) $
9. $ 4w^2(9w^3 + 10w^2 - 40w - 5) $
10. $ b(m + 10)(c - 5) $

Let me know if you'd like these explained in more detail or checked with substitution!