Factoring out the GCF worksheet with algebraic expressions.

To solve the problem of factoring out the greatest common factor (GCF) from each expression, we need to identify the largest factor that divides all terms in the expression. This includes both numerical coefficients and variable factors. Let's go through each expression step by step.

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Expression 1: \( 15x^3 - 25x^2 + 55x \)

1. Identify the GCF of the coefficients: The coefficients are \( 15, -25, \) and \( 55 \). The GCF of these numbers is \( 5 \).
2. Identify the GCF of the variables: The variables are \( x^3, x^2, \) and \( x \). The smallest power of \( x \) is \( x \). So, the GCF of the variables is \( x \).
3. Combine the GCF: The overall GCF is \( 5x \).
4. Factor out \( 5x \):
\[
15x^3 - 25x^2 + 55x = 5x(3x^2 - 5x + 11)
\]

Answer for Expression 1: \( 5x(3x^2 - 5x + 11) \)

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

1. Identify the GCF of the coefficients: The coefficients are \( 36, 63, \) and \( -27 \). The GCF of these numbers is \( 9 \).
2. Identify the GCF of the variables: The variables are \( x^3, x^2, \) and \( x \). The smallest power of \( x \) is \( x \). So, the GCF of the variables is \( x \).
3. Combine the GCF: The overall GCF is \( 9x \).
4. Factor out \( 9x \):
\[
36x^3 + 63x^2 - 27x = 9x(4x^2 + 7x - 3)
\]

Answer for Expression 2: \( 9x(4x^2 + 7x - 3) \)

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Expression 3: \( 25x^{12} - 35x^6 \)

1. Identify the GCF of the coefficients: The coefficients are \( 25 \) and \( -35 \). The GCF of these numbers is \( 5 \).
2. Identify the GCF of the variables: The variables are \( x^{12} \) and \( x^6 \). The smallest power of \( x \) is \( x^6 \). So, the GCF of the variables is \( x^6 \).
3. Combine the GCF: The overall GCF is \( 5x^6 \).
4. Factor out \( 5x^6 \):
\[
25x^{12} - 35x^6 = 5x^6(5x^6 - 7)
\]

Answer for Expression 3: \( 5x^6(5x^6 - 7) \)

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

1. Identify the GCF of the coefficients: The coefficients are \( -6 \) and \( -6 \). The GCF of these numbers is \( -6 \).
2. Identify the GCF of the variables: The variables are \( p^5 \) and \( p^4 \). The smallest power of \( p \) is \( p^4 \). So, the GCF of the variables is \( p^4 \).
3. Combine the GCF: The overall GCF is \( -6p^4 \).
4. Factor out \( -6p^4 \):
\[
-6p^5 - 6p^4 = -6p^4(p + 1)
\]

Answer for Expression 4: \( -6p^4(p + 1) \)

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

1. Identify the GCF of the coefficients: The coefficients are \( 72, -72, \) and \( -80 \). The GCF of these numbers is \( 8 \).
2. Identify the GCF of the variables: The variables are \( x^5, x^3, \) and \( x^2 \). The smallest power of \( x \) is \( x^2 \). So, the GCF of the variables is \( x^2 \).
3. Combine the GCF: The overall GCF is \( 8x^2 \).
4. Factor out \( 8x^2 \):
\[
72x^5 - 72x^3 - 80x^2 = 8x^2(9x^3 - 9x - 10)
\]

Answer for Expression 5: \( 8x^2(9x^3 - 9x - 10) \)

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

1. Identify the GCF of the coefficients: The coefficients are \( 6, 2, \) and \( -4 \). The GCF of these numbers is \( 2 \).
2. Identify the GCF of the variables: The variables are \( x^2yz, xy^2z, \) and \( xyz \). The smallest powers are \( x, y, \) and \( z \). So, the GCF of the variables is \( xyz \).
3. Combine the GCF: The overall GCF is \( 2xyz \).
4. Factor out \( 2xyz \):
\[
6x^2yz + 2xy^2z - 4xyz = 2xyz(3x + y - 2)
\]

Answer for Expression 6: \( 2xyz(3x + y - 2) \)

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Expression 7: \( -16p^3q^2 + 24p^2q^3 - 32p^4q \)

1. Identify the GCF of the coefficients: The coefficients are \( -16, 24, \) and \( -32 \). The GCF of these numbers is \( -8 \).
2. Identify the GCF of the variables: The variables are \( p^3q^2, p^2q^3, \) and \( p^4q \). The smallest powers are \( p^2 \) and \( q \). So, the GCF of the variables is \( p^2q \).
3. Combine the GCF: The overall GCF is \( -8p^2q \).
4. Factor out \( -8p^2q \):
\[
-16p^3q^2 + 24p^2q^3 - 32p^4q = -8p^2q(2pq - 3q^2 + 4p^2)
\]

Answer for Expression 7: \( -8p^2q(2pq - 3q^2 + 4p^2) \)

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Expression 8: \( 7wx(a - 9) - 10w(9 - a) \)

1. Simplify the second term: Notice that \( 9 - a = -(a - 9) \). So, the expression becomes:
\[
7wx(a - 9) - 10w(-(a - 9)) = 7wx(a - 9) + 10w(a - 9)
\]
2. Identify the GCF: The GCF of the terms is \( w(a - 9) \).
3. Factor out \( w(a - 9) \):
\[
7wx(a - 9) + 10w(a - 9) = w(a - 9)(7x + 10)
\]

Answer for Expression 8: \( w(a - 9)(7x + 10) \)

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Expression 9: \( 36w^5 + 40w^4 - 160w^3 - 20w^2 \)

1. Identify the GCF of the coefficients: The coefficients are \( 36, 40, -160, \) and \( -20 \). The GCF of these numbers is \( 4 \).
2. Identify the GCF of the variables: The variables are \( w^5, w^4, w^3, \) and \( w^2 \). The smallest power of \( w \) is \( w^2 \). So, the GCF of the variables is \( w^2 \).
3. Combine the GCF: The overall GCF is \( 4w^2 \).
4. Factor out \( 4w^2 \):
\[
36w^5 + 40w^4 - 160w^3 - 20w^2 = 4w^2(9w^3 + 10w^2 - 40w - 5)
\]

Answer for Expression 9: \( 4w^2(9w^3 + 10w^2 - 40w - 5) \)

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

1. Simplify the second term: Notice that \( 10 + m = m + 10 \). So, the expression becomes:
\[
bc(m + 10) - 5b(m + 10)
\]
2. Identify the GCF: The GCF of the terms is \( b(m + 10) \).
3. Factor out \( b(m + 10) \):
\[
bc(m + 10) - 5b(m + 10) = b(m + 10)(c - 5)
\]

Answer for Expression 10: \( b(m + 10)(c - 5) \)

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Final Answers

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