Factoring using GCF Worksheet | PDF Printable Algebra Worksheet - Free Printable

Problem: Factoring Using the Greatest Common Factor (GCF)

The task is to factorize the given expressions fully by identifying and extracting the greatest common factor (GCF) from each term. Below, I will solve a few examples from both Section A and Section B to illustrate the process.

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Section A: Factoring Expressions

#### 1) \( 8x + 24 \)

- Step 1: Identify the GCF of the coefficients \( 8 \) and \( 24 \).
- The factors of \( 8 \) are \( 1, 2, 4, 8 \).
- The factors of \( 24 \) are \( 1, 2, 3, 4, 6, 8, 12, 24 \).
- The greatest common factor is \( 8 \).

- Step 2: Factor out the GCF \( 8 \) from each term.
\[
8x + 24 = 8(x) + 8(3) = 8(x + 3)
\]

- Final Answer:
\[
\boxed{8(x + 3)}
\]

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#### 7) \( 21xy + 14x \)

- Step 1: Identify the GCF of the coefficients \( 21 \) and \( 14 \).
- The factors of \( 21 \) are \( 1, 3, 7, 21 \).
- The factors of \( 14 \) are \( 1, 2, 7, 14 \).
- The greatest common factor is \( 7 \).

- Step 2: Identify the common variable factors.
- Both terms have \( x \), so the common variable factor is \( x \).

- Step 3: Combine the GCF of coefficients and variables.
- The overall GCF is \( 7x \).

- Step 4: Factor out \( 7x \) from each term.
\[
21xy + 14x = 7x(3y) + 7x(2) = 7x(3y + 2)
\]

- Final Answer:
\[
\boxed{7x(3y + 2)}
\]

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#### 13) \( 9h^2g - 15h^3 \)

- Step 1: Identify the GCF of the coefficients \( 9 \) and \( 15 \).
- The factors of \( 9 \) are \( 1, 3, 9 \).
- The factors of \( 15 \) are \( 1, 3, 5, 15 \).
- The greatest common factor is \( 3 \).

- Step 2: Identify the common variable factors.
- Both terms have \( h^2 \) (since \( h^3 = h^2 \cdot h \)).
- Both terms have \( g \) in the first term but not in the second term, so the common variable factor is just \( h^2 \).

- Step 3: Combine the GCF of coefficients and variables.
- The overall GCF is \( 3h^2 \).

- Step 4: Factor out \( 3h^2 \) from each term.
\[
9h^2g - 15h^3 = 3h^2(3g) - 3h^2(5h) = 3h^2(3g - 5h)
\]

- Final Answer:
\[
\boxed{3h^2(3g - 5h)}
\]

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Section B: Factoring More Complex Expressions

#### 1) \( 6 - 12gh + 3h \)

- Step 1: Identify the GCF of the coefficients \( 6 \), \( -12 \), and \( 3 \).
- The factors of \( 6 \) are \( 1, 2, 3, 6 \).
- The factors of \( -12 \) are \( 1, 2, 3, 4, 6, 12 \) (ignoring the sign).
- The factors of \( 3 \) are \( 1, 3 \).
- The greatest common factor is \( 3 \).

- Step 2: Factor out the GCF \( 3 \) from each term.
\[
6 - 12gh + 3h = 3(2) - 3(4gh) + 3(h) = 3(2 - 4gh + h)
\]

- Final Answer:
\[
\boxed{3(2 - 4gh + h)}
\]

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#### 7) \( 9wu - 27wuv + 45w \)

- Step 1: Identify the GCF of the coefficients \( 9 \), \( -27 \), and \( 45 \).
- The factors of \( 9 \) are \( 1, 3, 9 \).
- The factors of \( -27 \) are \( 1, 3, 9, 27 \) (ignoring the sign).
- The factors of \( 45 \) are \( 1, 3, 5, 9, 15, 45 \).
- The greatest common factor is \( 9 \).

- Step 2: Identify the common variable factors.
- All terms have \( w \), so the common variable factor is \( w \).

- Step 3: Combine the GCF of coefficients and variables.
- The overall GCF is \( 9w \).

- Step 4: Factor out \( 9w \) from each term.
\[
9wu - 27wuv + 45w = 9w(u) - 9w(3uv) + 9w(5) = 9w(u - 3uv + 5)
\]

- Final Answer:
\[
\boxed{9w(u - 3uv + 5)}
\]

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#### 13) \( 7xyz + xy^2 - x^2y \)

- Step 1: Identify the GCF of the coefficients \( 7 \), \( 1 \), and \( -1 \).
- The greatest common factor is \( 1 \).

- Step 2: Identify the common variable factors.
- All terms have \( x \) and \( y \), so the common variable factor is \( xy \).

- Step 3: Combine the GCF of coefficients and variables.
- The overall GCF is \( xy \).

- Step 4: Factor out \( xy \) from each term.
\[
7xyz + xy^2 - x^2y = xy(7z) + xy(y) - xy(x) = xy(7z + y - x)
\]

- Final Answer:
\[
\boxed{xy(7z + y - x)}
\]

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General Approach Summary

1. Identify the GCF of the coefficients.
2. Identify the common variable factors.
3. Combine the GCF of coefficients and variables.
4. Factor out the GCF from each term.

This method can be applied to all the problems in the worksheet. If you need solutions for more specific problems, feel free to ask!

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Final Boxed Answers for Selected Problems:
- Section A:
- \( 1) \boxed{8(x + 3)} \)
- \( 7) \boxed{7x(3y + 2)} \)
- \( 13) \boxed{3h^2(3g - 5h)} \)

- Section B:
- \( 1) \boxed{3(2 - 4gh + h)} \)
- \( 7) \boxed{9w(u - 3uv + 5)} \)
- \( 13) \boxed{xy(7z + y - x)} \)