Factoring Polynomials Worksheets with Answer Key - Free Printable

Absolutely! Let’s solve each problem by factoring out the Greatest Common Factor (GCF). The GCF is the largest expression that divides evenly into all terms of the polynomial.

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Problem 1: `15x³ - 25x² + 55x`

- Coefficients: 15, 25, 55 → GCF = 5
- Variables: x³, x², x → lowest power = x
- GCF = 5x

Factor out 5x:

> 5x(3x² - 5x + 11)

✔ Final Answer: 5x(3x² - 5x + 11)

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Problem 2: `36x³ + 63x² - 27x`

- Coefficients: 36, 63, 27 → GCF = 9
- Variables: x³, x², x → lowest power = x
- GCF = 9x

Factor out 9x:

> 9x(4x² + 7x - 3)

✔ Final Answer: 9x(4x² + 7x - 3)

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Problem 3: `25x¹² - 35x⁶`

- Coefficients: 25, 35 → GCF = 5
- Variables: x¹², x⁶ → lowest power = x⁶
- GCF = 5x⁶

Factor out 5x⁶:

> 5x⁶(5x⁶ - 7)

✔ Final Answer: 5x⁶(5x⁶ - 7)

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Problem 4: `-6p⁵ - 6p⁴`

- Coefficients: -6, -6 → GCF = -6 (we can factor out negative to make leading term positive if desired, but technically GCF is 6; however, factoring out -6 is common to keep first term positive inside)
- Variables: p⁵, p⁴ → lowest power = p⁴
- GCF = -6p⁴ (or 6p⁴ — we’ll use -6p⁴ to make inside expression start with positive)

Factor out -6p⁴:

> -6p⁴(p + 1)

✔ Final Answer: -6p⁴(p + 1)
*(Alternatively: 6p⁴(-p - 1), but the first form is standard)*

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Problem 5: `72x⁵ - 72x³ - 80x²`

- Coefficients: 72, 72, 80 → GCF = 8
- Variables: x⁵, x³, x² → lowest power = x²
- GCF = 8x²

Factor out 8x²:

> 8x²(9x³ - 9x - 10)

✔ Final Answer: 8x²(9x³ - 9x - 10)

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Problem 6: `6x²yz + 2xy²z - 4xyz`

- Coefficients: 6, 2, 4 → GCF = 2
- Variables: All have x, y, z → lowest powers: x¹, y¹, z¹ → xyz
- GCF = 2xyz

Factor out 2xyz:

> 2xyz(3x + y - 2)

✔ Final Answer: 2xyz(3x + y - 2)

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Problem 7: `-16p³q² + 24p²q³ - 32p⁴q`

- Coefficients: -16, 24, -32 → GCF = 8 (but since first term is negative, often factor out -8 to make inside start positive)
- Variables: p³q², p²q³, p⁴q → lowest powers: p², q¹ → p²q
- GCF = -8p²q

Factor out -8p²q:

> -8p²q(2p q - 3q² + 4p²)
Wait — let’s do it carefully:

Original:
-16p³q² + 24p²q³ - 32p⁴q
= -8p²q * (2p q) → gives -16p³q² ✔
= -8p²q * (-3q²) → gives +24p²q³ ✔
= -8p²q * (4p²) → gives -32p⁴q ✔

So:
> -8p²q(2pq - 3q² + 4p²)

But usually we write in order of descending powers of p:

> -8p²q(4p² + 2pq - 3q²)

✔ Final Answer: -8p²q(4p² + 2pq - 3q²)

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Problem 8: `7wx(a - 9) - 10w(9 - a)`

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

So rewrite second term:

> `7wx(a - 9) - 10w(-(a - 9))`
> = `7wx(a - 9) + 10w(a - 9)`

Now both terms have common factor: w(a - 9)

Factor that out:

> w(a - 9)(7x + 10)

✔ Final Answer: w(a - 9)(7x + 10)

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Problem 9: `36w⁵ + 40w⁴ - 160w³ - 20w²`

- Coefficients: 36, 40, 160, 20 → GCF = 4
- Variables: w⁵, w⁴, w³, w² → lowest power = w²
- GCF = 4w²

Factor out 4w²:

> 4w²(9w³ + 10w² - 40w - 5)

✔ Final Answer: 4w²(9w³ + 10w² - 40w - 5)

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Problem 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 out:

> b(m + 10)(c - 5)

✔ Final Answer: b(m + 10)(c - 5)

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## ✔ FINAL ANSWERS SUMMARY:

1. 5x(3x² - 5x + 11)
2. 9x(4x² + 7x - 3)
3. 5x⁶(5x⁶ - 7)
4. -6p⁴(p + 1)
5. 8x²(9x³ - 9x - 10)
6. 2xyz(3x + y - 2)
7. -8p²q(4p² + 2pq - 3q²)
8. w(a - 9)(7x + 10)
9. 4w²(9w³ + 10w² - 40w - 5)
10. b(m + 10)(c - 5)

Let me know if you’d like step-by-step breakdowns for any specific one!