Factoring using GCF worksheet with algebraic expressions for practice.

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 variables in both terms.
- Both terms have \( x \), so the variable part of the GCF is \( x \).

- Step 3: Combine the numerical and variable parts of the GCF.
- The 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 variables in both terms.
- Both terms have \( h^2 \) (since \( h^3 = h^2 \cdot h \)).
- The variable part of the GCF is \( h^2 \).

- Step 3: Combine the numerical and variable parts of the GCF.
- The 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 variables in all terms.
- All terms have \( w \), so the variable part of the GCF is \( w \).

- Step 3: Combine the numerical and variable parts of the GCF.
- The 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 variables in all terms.
- All terms have \( x \) and \( y \), so the variable part of the GCF is \( xy \).

- Step 3: Combine the numerical and variable parts of the GCF.
- The 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 variables in all terms.
3. Combine the numerical and variable parts to form the GCF.
4. Factor out the GCF from each term.
5. Write the expression as the product of the GCF and the remaining terms in parentheses.

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

Final Boxed Answers for Examples:
\[
\boxed{8(x + 3)}, \quad \boxed{7x(3y + 2)}, \quad \boxed{3h^2(3g - 5h)}, \quad \boxed{3(2 - 4gh + h)}, \quad \boxed{9w(u - 3uv + 5)}, \quad \boxed{xy(7z + y - x)}
\]