Factoring Polynomials Worksheets with Answer Key - Free Printable

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.

---

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

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

Factor it out:
$$
= 5x(3x^2 - 5x + 11)
$$

✔ Answer: $ \boxed{5x(3x^2 - 5x + 11)} $

---

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

- Coefficients: 36, 63, 27 → GCF is 9
- Variables: All have at least $ x $ → GCF is $ x $
- GCF = $ 9x $

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

✔ Answer: $ \boxed{9x(4x^2 + 7x - 3)} $

---

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

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

Factor:
$$
= 5x^6(5x^6 - 7)
$$

✔ Answer: $ \boxed{5x^6(5x^6 - 7)} $

---

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

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

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

✔ Answer: $ \boxed{-6p^4(p + 1)} $

---

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

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

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

✔ Answer: $ \boxed{8x^2(9x^3 - 9x - 10)} $

---

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

- Coefficients: 6, 2, 4 → GCF is 2
- Variables: All terms have $ x, y, z $ → GCF is $ xyz $
- GCF = $ 2xyz $

Factor:
$$
= 2xyz(3x + y - 2)
$$

✔ Answer: $ \boxed{2xyz(3x + y - 2)} $

---

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

- Coefficients: 16, 24, 32 → GCF is 8
- But signs: first term negative → factor out -8
- Variables: $ p^2q $ is common in all → GCF = $ -8p^2q $

Now divide each term:

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

So:
$$
= -8p^2q(2pq - 3q^2 + 4p^2)
$$

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

---

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

Notice: $ (a - 9) $ and $ (9 - a) $ are opposites:
$$
(9 - a) = -(a - 9)
$$

So rewrite:
$$
= 7wx(a - 9) - 10w(-1)(a - 9)
= 7wx(a - 9) + 10w(a - 9)
$$

Now both terms have $ w(a - 9) $ as common factor:
$$
= w(a - 9)(7x + 10)
$$

✔ Answer: $ \boxed{w(a - 9)(7x + 10)} $

---

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

- Coefficients: 36, 40, 160, 20 → GCF? Let's find:
- GCF of 36, 40, 160, 20 → smallest is 4
- Variables: all have $ w^2 $ → GCF = $ 4w^2 $

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

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

---

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

Note: $ (m + 10) = (10 + m) $ → same thing

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

Common factor: $ b(m + 10) $

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

✔ Answer: $ \boxed{b(m + 10)(c - 5)} $

---

✔ 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 a printed version or explanations on how to find GCFs!