Factoring out the GCF math worksheet with ten algebraic expressions.

Sure! Let’s solve each problem step by step. The goal is to factor out the Greatest Common Factor (GCF) from each expression.

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Problem 1:

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

- Coefficients: 15, 25, 55 → GCF = 5
- Variables: All terms 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: \( x^{12} \) and \( x^6 \) → GCF = \( x^6 \)
- 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 (we can factor out -6 to make leading term positive if preferred, but technically GCF is 6; however, factoring out -6 is common to avoid leading negative)
- Variables: \( p^5 \) and \( p^4 \) → GCF = \( p^4 \)
- So, GCF = \( -6p^4 \) (to make inside positive)

Factor it out:
\[
-6p^4(p + 1)
\]

✔ Answer: \( \boxed{-6p^4(p + 1)} \)

*(Note: You could also write \( 6p^4(-p -1) \), but the first form is more standard.)*

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Problem 5:

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

- Coefficients: 72, 72, 80 → GCF = 8
- Variables: \( x^5, x^3, x^2 \) → GCF = \( x^2 \)
- 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: Each term has at least \( x, y, z \) → GCF = \( xyz \)
- 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
- Variables:
- \( p^3, p^2, p^4 \) → min exponent = \( p^2 \)
- \( q^2, q^3, q^1 \) → min exponent = \( q^1 \)
- So, GCF = \( 8p^2q \) — but since first term is negative, we’ll factor out -8p^2q to make leading coefficient positive inside.

Factor it out:
\[
-8p^2q(2p - 3q + 4p^2)
\]
Wait — let’s check signs:

Original:
\( -16p^3q^2 = -8p^2q \cdot 2p q \)
\( +24p^2q^3 = -8p^2q \cdot (-3q^2) \)
\( -32p^4q = -8p^2q \cdot 4p^2 \)

So inside: \( 2pq - 3q^2 + 4p^2 \) — but order matters for clarity.

Better to write as:
\[
-8p^2q(2pq - 3q^2 + 4p^2)
\]
Or rearrange terms:
\[
-8p^2q(4p^2 + 2pq - 3q^2)
\]

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

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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 \( w(a - 9) \)

Factor that 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: \( w^5, w^4, w^3, w^2 \) → GCF = \( w^2 \)
- 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. \( \boxed{-8p^2q(4p^2 + 2pq - 3q^2)} \)
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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