Perfect Cubes and Factoring

Identifying Perfect Cubes

• A perfect cube is a number that can be expressed as the cube of an integer.
• Examples:
  • 8: Yes, because (2^3 = 2 \times 2 \times 2 = 8)
  • 25: No, it's a perfect square, not a perfect cube.
  • 64: Yes, because (4^3 = 4 \times 4 \times 4 = 64)
  • 40: No, not a perfect cube.
  • 27: Yes, because (3^3 = 3 \times 3 \times 3 = 27)
  • 60: No, not a perfect cube.
  • 125: Yes, because (5^3 = 5 \times 5 \times 5 = 125)
  • 72: No, not a perfect cube.
  • 216: Yes, because (6^3 = 6 \times 6 \times 6 = 216)
  • 343: Yes, because (7^3 = 7 \times 7 \times 7 = 343)

Evaluating Exponential Notation

• Exponential evaluations of cubes:
  • (2^3 = 8)
  • (3^3 = 27)
  • (4^3 = 64)
  • (5^3 = 125)
  • (6^3 = 216)
  • (7^3 = 343)

Factoring the Sum of Two Cubes

• Formula: (x^3 + y^3 = (x + y)(x^2 - xy + y^2))

• Example: Factor (a^3 + 64)

  • Rewrite as (a^3 + 4^3)
  • Apply the formula: ( (a + 4)(a^2 - 4a + 16) )

• Example: Factor (8b^3 + 27c^3)

  • Rewrite as ((2b)^3 + (3c)^3)
  • Apply the formula: ( (2b + 3c)(4b^2 - 6bc + 9c^2) )

Factoring the Difference of Two Cubes

• Formula: (x^3 - y^3 = (x - y)(x^2 + xy + y^2))

• Example: Factor (27c^3 - d^3)

  • Rewrite as ((3c)^3 - d^3)
  • Apply the formula: ( (3c - d)(9c^2 + 3cd + d^2) )

• Example: Factor (8e^6f^6 - 125g^3)

  • Rewrite as ((2ef^2)^3 - (5g)^3)
  • Apply the formula: ( (2ef^2 - 5g)(4e^2f^4 + 10ef^2g + 25g^2) )

• Example: Factor (64 - p^6)

  • Rewrite as (4^3 - (p^2)^3)
  • Apply the formula: ( (4 - p^2)(16 + 4p^2 + p^4) )

Conclusion

• Understanding perfect cubes and how to factor sums and differences of cubes is crucial for solving polynomial equations.
• Remember the formulas and practice with examples for better retention.