Worksheet #2 for factoring polynomials using the greatest common factor.

To solve the problem of factoring the given polynomials using the greatest common factor (GCF), we will follow these steps:

1. Identify the GCF of all terms in the polynomial.
2. Factor out the GCF from each term.
3. Write the polynomial as the product of the GCF and the remaining expression.

Let's go through each problem step by step.

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Problem 1: \( -16x^2 + 20 \)

- Step 1: Identify the GCF of the coefficients \(-16\) and \(20\).
- The factors of \(-16\) are: \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16\).
- The factors of \(20\) are: \(\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\).
- The greatest common factor of \(-16\) and \(20\) is \(4\).

- Step 2: Factor out \(4\) from each term.
- \( -16x^2 = 4 \cdot (-4x^2) \)
- \( 20 = 4 \cdot 5 \)

- Step 3: Write the polynomial as:
\[
-16x^2 + 20 = 4(-4x^2 + 5)
\]

Answer:
\[
\boxed{4(-4x^2 + 5)}
\]

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Problem 2: \( -4n^3 - n \)

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

- Step 2: Identify the common variable factor.
- Both terms have \(n\) as a common factor.
- The smallest power of \(n\) is \(n^1\).

- Step 3: Factor out \(n\) from each term.
- \( -4n^3 = n \cdot (-4n^2) \)
- \( -n = n \cdot (-1) \)

- Step 4: Write the polynomial as:
\[
-4n^3 - n = n(-4n^2 - 1)
\]

Answer:
\[
\boxed{n(-4n^2 - 1)}
\]

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Problem 3: \( 16x + 20 \)

- Step 1: Identify the GCF of the coefficients \(16\) and \(20\).
- The factors of \(16\) are: \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16\).
- The factors of \(20\) are: \(\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\).
- The greatest common factor of \(16\) and \(20\) is \(4\).

- Step 2: Factor out \(4\) from each term.
- \( 16x = 4 \cdot 4x \)
- \( 20 = 4 \cdot 5 \)

- Step 3: Write the polynomial as:
\[
16x + 20 = 4(4x + 5)
\]

Answer:
\[
\boxed{4(4x + 5)}
\]

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Problem 4: \( -3x^2 - 3x \)

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

- Step 2: Identify the common variable factor.
- Both terms have \(x\) as a common factor.
- The smallest power of \(x\) is \(x^1\).

- Step 3: Factor out \(3x\) from each term.
- \( -3x^2 = 3x \cdot (-x) \)
- \( -3x = 3x \cdot (-1) \)

- Step 4: Write the polynomial as:
\[
-3x^2 - 3x = 3x(-x - 1)
\]

Answer:
\[
\boxed{3x(-x - 1)}
\]

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Problem 5: \( -20x + 15 \)

- Step 1: Identify the GCF of the coefficients \(-20\) and \(15\).
- The factors of \(-20\) are: \(\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\).
- The factors of \(15\) are: \(\pm 1, \pm 3, \pm 5, \pm 15\).
- The greatest common factor of \(-20\) and \(15\) is \(5\).

- Step 2: Factor out \(5\) from each term.
- \( -20x = 5 \cdot (-4x) \)
- \( 15 = 5 \cdot 3 \)

- Step 3: Write the polynomial as:
\[
-20x + 15 = 5(-4x + 3)
\]

Answer:
\[
\boxed{5(-4x + 3)}
\]

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Problem 6: \( 5x - 10x^3 \)

- Step 1: Identify the GCF of the coefficients \(5\) and \(-10\).
- The factors of \(5\) are: \(\pm 1, \pm 5\).
- The factors of \(-10\) are: \(\pm 1, \pm 2, \pm 5, \pm 10\).
- The greatest common factor of \(5\) and \(-10\) is \(5\).

- Step 2: Identify the common variable factor.
- Both terms have \(x\) as a common factor.
- The smallest power of \(x\) is \(x^1\).

- Step 3: Factor out \(5x\) from each term.
- \( 5x = 5x \cdot 1 \)
- \( -10x^3 = 5x \cdot (-2x^2) \)

- Step 4: Write the polynomial as:
\[
5x - 10x^3 = 5x(1 - 2x^2)
\]

Answer:
\[
\boxed{5x(1 - 2x^2)}
\]

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Problem 7: \( 9a - 6a^2 \)

- Step 1: Identify the GCF of the coefficients \(9\) and \(-6\).
- The factors of \(9\) are: \(\pm 1, \pm 3, \pm 9\).
- The factors of \(-6\) are: \(\pm 1, \pm 2, \pm 3, \pm 6\).
- The greatest common factor of \(9\) and \(-6\) is \(3\).

- Step 2: Identify the common variable factor.
- Both terms have \(a\) as a common factor.
- The smallest power of \(a\) is \(a^1\).

- Step 3: Factor out \(3a\) from each term.
- \( 9a = 3a \cdot 3 \)
- \( -6a^2 = 3a \cdot (-2a) \)

- Step 4: Write the polynomial as:
\[
9a - 6a^2 = 3a(3 - 2a)
\]

Answer:
\[
\boxed{3a(3 - 2a)}
\]

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Problem 8: \( -5x^7 - x^3 \)

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

- Step 2: Identify the common variable factor.
- Both terms have \(x\) as a common factor.
- The smallest power of \(x\) is \(x^3\).

- Step 3: Factor out \(x^3\) from each term.
- \( -5x^7 = x^3 \cdot (-5x^4) \)
- \( -x^3 = x^3 \cdot (-1) \)

- Step 4: Write the polynomial as:
\[
-5x^7 - x^3 = x^3(-5x^4 - 1)
\]

Answer:
\[
\boxed{x^3(-5x^4 - 1)}
\]

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Problem 9: \( 5x + 5 \)

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

- Step 2: Factor out \(5\) from each term.
- \( 5x = 5 \cdot x \)
- \( 5 = 5 \cdot 1 \)

- Step 3: Write the polynomial as:
\[
5x + 5 = 5(x + 1)
\]

Answer:
\[
\boxed{5(x + 1)}
\]

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Problem 10: \( -x^2 - 2x \)

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

- Step 2: Identify the common variable factor.
- Both terms have \(x\) as a common factor.
- The smallest power of \(x\) is \(x^1\).

- Step 3: Factor out \(x\) from each term.
- \( -x^2 = x \cdot (-x) \)
- \( -2x = x \cdot (-2) \)

- Step 4: Write the polynomial as:
\[
-x^2 - 2x = x(-x - 2)
\]

Answer:
\[
\boxed{x(-x - 2)}
\]

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Final Answers:

1. \(\boxed{4(-4x^2 + 5)}\)
2. \(\boxed{n(-4n^2 - 1)}\)
3. \(\boxed{4(4x + 5)}\)
4. \(\boxed{3x(-x - 1)}\)
5. \(\boxed{5(-4x + 3)}\)
6. \(\boxed{5x(1 - 2x^2)}\)
7. \(\boxed{3a(3 - 2a)}\)
8. \(\boxed{x^3(-5x^4 - 1)}\)
9. \(\boxed{5(x + 1)}\)
10. \(\boxed{x(-x - 2)}\)