Day 3 Factoring Cubics | PDF | Algebra | Equations - Free Printable

Problem Overview:

The task involves factoring cubic equations completely. We will solve each problem step by step, using techniques such as grouping, the Rational Root Theorem, synthetic division, and recognizing special forms like sums and differences of cubes.

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Part 1: Factoring Each Completely

#### Problem 1: \( 16r^3 - 6r^2 - 56r + 21 \)

1. Group terms:
\[
(16r^3 - 6r^2) + (-56r + 21)
\]

2. Factor out the GCF from each group:
\[
2r^2(8r - 3) - 7(8r - 3)
\]

3. Factor out the common binomial factor:
\[
(8r - 3)(2r^2 - 7)
\]

Final Answer: \(\boxed{(8r - 3)(2r^2 - 7)}\)

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#### Problem 2: \( 42x^3 + 24x^2 + 49x + 28 \)

1. Group terms:
\[
(42x^3 + 24x^2) + (49x + 28)
\]

2. Factor out the GCF from each group:
\[
6x^2(7x + 4) + 7(7x + 4)
\]

3. Factor out the common binomial factor:
\[
(7x + 4)(6x^2 + 7)
\]

Final Answer: \(\boxed{(7x + 4)(6x^2 + 7)}\)

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#### Problem 3: \( 21n^3 - 3n^2 - 35n + 5 \)

1. Group terms:
\[
(21n^3 - 3n^2) + (-35n + 5)
\]

2. Factor out the GCF from each group:
\[
3n^2(7n - 1) - 5(7n - 1)
\]

3. Factor out the common binomial factor:
\[
(7n - 1)(3n^2 - 5)
\]

Final Answer: \(\boxed{(7n - 1)(3n^2 - 5)}\)

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#### Problem 4: \( 42b^3 - 24b^2 - 35b + 20 \)

1. Group terms:
\[
(42b^3 - 24b^2) + (-35b + 20)
\]

2. Factor out the GCF from each group:
\[
6b^2(7b - 4) - 5(7b - 4)
\]

3. Factor out the common binomial factor:
\[
(7b - 4)(6b^2 - 5)
\]

Final Answer: \(\boxed{(7b - 4)(6b^2 - 5)}\)

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#### Problem 5: \( 40x^3 - 48x^2 - 25x + 30 \)

1. Group terms:
\[
(40x^3 - 48x^2) + (-25x + 30)
\]

2. Factor out the GCF from each group:
\[
8x^2(5x - 6) - 5(5x - 6)
\]

3. Factor out the common binomial factor:
\[
(5x - 6)(8x^2 - 5)
\]

Final Answer: \(\boxed{(5x - 6)(8x^2 - 5)}\)

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#### Problem 6: \( 40v^3 - 15v^2 - 16v + 6 \)

1. Group terms:
\[
(40v^3 - 15v^2) + (-16v + 6)
\]

2. Factor out the GCF from each group:
\[
5v^2(8v - 3) - 2(8v - 3)
\]

3. Factor out the common binomial factor:
\[
(8v - 3)(5v^2 - 2)
\]

Final Answer: \(\boxed{(8v - 3)(5v^2 - 2)}\)

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#### Problem 7: \( 10n^3 - 2n^2 - 25n + 5 \)

1. Group terms:
\[
(10n^3 - 2n^2) + (-25n + 5)
\]

2. Factor out the GCF from each group:
\[
2n^2(5n - 1) - 5(5n - 1)
\]

3. Factor out the common binomial factor:
\[
(5n - 1)(2n^2 - 5)
\]

Final Answer: \(\boxed{(5n - 1)(2n^2 - 5)}\)

---

#### Problem 8: \( 5v^3 - 30v^2 + 2v - 12 \)

1. Group terms:
\[
(5v^3 - 30v^2) + (2v - 12)
\]

2. Factor out the GCF from each group:
\[
5v^2(v - 6) + 2(v - 6)
\]

3. Factor out the common binomial factor:
\[
(v - 6)(5v^2 + 2)
\]

Final Answer: \(\boxed{(v - 6)(5v^2 + 2)}\)

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#### Problem 9: \( 3a^3 - 7a^2 - 9a + 21 \)

1. Group terms:
\[
(3a^3 - 7a^2) + (-9a + 21)
\]

2. Factor out the GCF from each group:
\[
a^2(3a - 7) - 3(3a - 7)
\]

3. Factor out the common binomial factor:
\[
(3a - 7)(a^2 - 3)
\]

Final Answer: \(\boxed{(3a - 7)(a^2 - 3)}\)

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#### Problem 10: \( 2x^3 + 6x^2 + 5x + 15 \)

1. Group terms:
\[
(2x^3 + 6x^2) + (5x + 15)
\]

2. Factor out the GCF from each group:
\[
2x^2(x + 3) + 5(x + 3)
\]

3. Factor out the common binomial factor:
\[
(x + 3)(2x^2 + 5)
\]

Final Answer: \(\boxed{(x + 3)(2x^2 + 5)}\)

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Part 2: Factor Each

#### Problem 11: \( x^3 + 8 = 0 \)

1. Recognize the sum of cubes:
\[
x^3 + 8 = x^3 + 2^3
\]

2. Use the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = x \) and \( b = 2 \):
\[
x^3 + 8 = (x + 2)(x^2 - 2x + 4)
\]

Final Answer: \(\boxed{(x + 2)(x^2 - 2x + 4)}\)

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#### Problem 12: \( x^3 + 27 = 0 \)

1. Recognize the sum of cubes:
\[
x^3 + 27 = x^3 + 3^3
\]

2. Use the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = x \) and \( b = 3 \):
\[
x^3 + 27 = (x + 3)(x^2 - 3x + 9)
\]

Final Answer: \(\boxed{(x + 3)(x^2 - 3x + 9)}\)

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#### Problem 13: \( x^3 - 1 = 0 \)

1. Recognize the difference of cubes:
\[
x^3 - 1 = x^3 - 1^3
\]

2. Use the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = x \) and \( b = 1 \):
\[
x^3 - 1 = (x - 1)(x^2 + x + 1)
\]

Final Answer: \(\boxed{(x - 1)(x^2 + x + 1)}\)

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#### Problem 14: \( x^3 - 125 = 0 \)

1. Recognize the difference of cubes:
\[
x^3 - 125 = x^3 - 5^3
\]

2. Use the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = x \) and \( b = 5 \):
\[
x^3 - 125 = (x - 5)(x^2 + 5x + 25)
\]

Final Answer: \(\boxed{(x - 5)(x^2 + 5x + 25)}\)

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#### Problem 15: \( 27x^3 - 8 = 0 \)

1. Recognize the difference of cubes:
\[
27x^3 - 8 = (3x)^3 - 2^3
\]

2. Use the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = 3x \) and \( b = 2 \):
\[
27x^3 - 8 = (3x - 2)((3x)^2 + (3x)(2) + 2^2)
\]
Simplify:
\[
27x^3 - 8 = (3x - 2)(9x^2 + 6x + 4)
\]

Final Answer: \(\boxed{(3x - 2)(9x^2 + 6x + 4)}\)

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#### Problem 16: \( 8x^3 + 27 = 0 \)

1. Recognize the sum of cubes:
\[
8x^3 + 27 = (2x)^3 + 3^3
\]

2. Use the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = 2x \) and \( b = 3 \):
\[
8x^3 + 27 = (2x + 3)((2x)^2 - (2x)(3) + 3^2)
\]
Simplify:
\[
8x^3 + 27 = (2x + 3)(4x^2 - 6x + 9)
\]

Final Answer: \(\boxed{(2x + 3)(4x^2 - 6x + 9)}\)

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#### Problem 17: \( 27x^3 - 1 = 0 \)

1. Recognize the difference of cubes:
\[
27x^3 - 1 = (3x)^3 - 1^3
\]

2. Use the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = 3x \) and \( b = 1 \):
\[
27x^3 - 1 = (3x - 1)((3x)^2 + (3x)(1) + 1^2)
\]
Simplify:
\[
27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1)
\]

Final Answer: \(\boxed{(3x - 1)(9x^2 + 3x + 1)}\)

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#### Problem 18: \( x^3 - 64 = 0 \)

1. Recognize the difference of cubes:
\[
x^3 - 64 = x^3 - 4^3
\]

2. Use the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = x \) and \( b = 4 \):
\[
x^3 - 64 = (x - 4)(x^2 + 4x + 16)
\]

Final Answer: \(\boxed{(x - 4)(x^2 + 4x + 16)}\)

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#### Problem 19: \( 27x^3 - 125 = 0 \)

1. Recognize the difference of cubes:
\[
27x^3 - 125 = (3x)^3 - 5^3
\]

2. Use the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = 3x \) and \( b = 5 \):
\[
27x^3 - 125 = (3x - 5)((3x)^2 + (3x)(5) + 5^2)
\]
Simplify:
\[
27x^3 - 125 = (3x - 5)(9x^2 + 15x + 25)
\]

Final Answer: \(\boxed{(3x - 5)(9x^2 + 15x + 25)}\)

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#### Problem 20: \( -27x^3 + 8 = 0 \)

1. Rewrite the equation:
\[
-27x^3 + 8 = 8 - 27x^3
\]

2. Recognize the difference of cubes:
\[
8 - 27x^3 = 2^3 - (3x)^3
\]

3. Use the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = 2 \) and \( b = 3x \):
\[
8 - 27x^3 = (2 - 3x)(2^2 + (2)(3x) + (3x)^2)
\]
Simplify:
\[
8 - 27x^3 = (2 - 3x)(4 + 6x + 9x^2)
\]

Final Answer: \(\boxed{(2 - 3x)(9x^2 + 6x + 4)}\)

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Final Answers:

1. \(\boxed{(8r - 3)(2r^2 - 7)}\)
2. \(\boxed{(7x + 4)(6x^2 + 7)}\)
3. \(\boxed{(7n - 1)(3n^2 - 5)}\)
4. \(\boxed{(7b - 4)(6b^2 - 5)}\)
5. \(\boxed{(5x - 6)(8x^2 - 5)}\)
6. \(\boxed{(8v - 3)(5v^2 - 2)}\)
7. \(\boxed{(5n - 1)(2n^2 - 5)}\)
8. \(\boxed{(v - 6)(5v^2 + 2)}\)
9. \(\boxed{(3a - 7)(a^2 - 3)}\)
10. \(\boxed{(x + 3)(2x^2 + 5)}\)
11. \(\boxed{(x + 2)(x^2 - 2x + 4)}\)
12. \(\boxed{(x + 3)(x^2 - 3x + 9)}\)
13. \(\boxed{(x - 1)(x^2 + x + 1)}\)
14. \(\boxed{(x - 5)(x^2 + 5x + 25)}\)
15. \(\boxed{(3x - 2)(9x^2 + 6x + 4)}\)
16. \(\boxed{(2x + 3)(4x^2 - 6x + 9)}\)
17. \(\boxed{(3x - 1)(9x^2 + 3x + 1)}\)
18. \(\boxed{(x - 4)(x^2 + 4x + 16)}\)
19. \(\boxed{(3x - 5)(9x^2 + 15x + 25)}\)
20. \(\boxed{(2 - 3x)(9x^2 + 6x + 4)}\)