Practice simplifying algebraic expressions with this dedicated factoring using GCF worksheet.

Problem: Factoring Using the Greatest Common Factor (GCF)

The task is to factorize the given expressions fully by identifying and factoring out the greatest common factor (GCF) from each term in the expression. 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 Using GCF

#### 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 \( h^3 = h^2 \cdot h \)), so 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 Using GCF

#### 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: \( 1, 2, 3, 4, 6, 12 \)
- 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: \( 1, 3, 9, 27 \)
- 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 Explanation

To factor an expression using the GCF:
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 in the expression.
5. Write the expression as the product of the GCF and the remaining terms in parentheses.

This method ensures that the expression is factored completely.

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

If you need solutions for more problems, feel free to ask!