Factoring Polynomials Worksheets with Answer Key - Free Printable

Sure! Let’s solve each problem by factoring out the Greatest Common Factor (GCF) from each expression. The GCF is the largest expression that divides all terms in the polynomial.

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Problem 1:

Expression: \( 15x^3 - 25x^2 + 55x \)

- Coefficients: 15, 25, 55 → GCF = 5
- Variables: All have at least \( x^1 \) → GCF = \( x \)
- So, GCF = \( 5x \)

Factor it out:

\[
5x(3x^2 - 5x + 11)
\]

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

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Problem 2:

Expression: \( 36x^3 + 63x^2 - 27x \)

- Coefficients: 36, 63, 27 → GCF = 9
- Variables: All have at least \( x \) → GCF = \( x \)
- So, GCF = \( 9x \)

Factor it out:

\[
9x(4x^2 + 7x - 3)
\]

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

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Problem 3:

Expression: \( 25x^{12} - 35x^6 \)

- Coefficients: 25, 35 → GCF = 5
- Variables: Lowest power of \( x \) is \( x^6 \)
- So, GCF = \( 5x^6 \)

Factor it out:

\[
5x^6(5x^6 - 7)
\]

✔ Answer: \( \boxed{5x^6(5x^6 - 7)} \)

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Problem 4:

Expression: \( -6p^5 - 6p^4 \)

- Coefficients: -6 and -6 → GCF = 6 (but we can factor out -6 to make leading term positive if desired — often preferred)
- Variables: Lowest power = \( p^4 \)
- So, GCF = \( -6p^4 \) (or 6p⁴ — both work; let’s use -6p⁴ for cleaner result)

Factor out \( -6p^4 \):

\[
-6p^4(p + 1)
\]

✔ Answer: \( \boxed{-6p^4(p + 1)} \)

*(Note: Factoring out 6p⁴ gives \( 6p^4(-p -1) \), which is equivalent.)*

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Problem 5:

Expression: \( 72x^5 - 72x^3 - 80x^2 \)

- Coefficients: 72, 72, 80 → GCF = 8
- Variables: Lowest power = \( x^2 \)
- So, GCF = \( 8x^2 \)

Factor it out:

\[
8x^2(9x^3 - 9x - 10)
\]

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

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Problem 6:

Expression: \( 6x^2yz + 2xy^2z - 4xyz \)

- Coefficients: 6, 2, 4 → GCF = 2
- Variables: All terms have \( x, y, z \) → lowest powers: \( x^1, y^1, z^1 \)
- So, GCF = \( 2xyz \)

Factor it out:

\[
2xyz(3x + y - 2)
\]

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

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Problem 7:

Expression: \( -16p^3q^2 + 24p^2q^3 - 32p^4q \)

- Coefficients: -16, 24, -32 → GCF = 8 (but since first term is negative, we can factor out -8 to make leading coefficient positive)
- Variables:
- p: min exponent = 2 → \( p^2 \)
- q: min exponent = 1 → \( q \)
- So, GCF = \( -8p^2q \)

Factor it out:

\[
-8p^2q(2pq - 3q^2 + 4p^2)
\]

*(Check: -8p²q × 2pq = -16p³q² ✔️
-8p²q × (-3q²) = +24p²q³ ✔️
-8p²q × 4p² = -32p⁴q ✔️)*

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

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Problem 8:

Expression: \( 7wx(a - 9) - 10w(9 - a) \)

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

So rewrite:

\[
7wx(a - 9) - 10w \cdot [-(a - 9)] = 7wx(a - 9) + 10w(a - 9)
\]

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

Factor it out:

\[
w(a - 9)(7x + 10)
\]

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

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Problem 9:

Expression: \( 36w^5 + 40w^4 - 160w^3 - 20w^2 \)

- Coefficients: 36, 40, 160, 20 → GCF = 4
- Variables: Lowest power = \( w^2 \)
- So, GCF = \( 4w^2 \)

Factor it out:

\[
4w^2(9w^3 + 10w^2 - 40w - 5)
\]

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

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Problem 10:

Expression: \( bc(m + 10) - 5b(10 + m) \)

Note: \( (m + 10) = (10 + m) \), so same binomial!

Rewrite:

\[
bc(m + 10) - 5b(m + 10)
\]

Common factor: \( b(m + 10) \)

Factor it out:

\[
b(m + 10)(c - 5)
\]

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

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## ✔ Final Answers Summary:

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

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