FACTORING EXPRESSIONS WITH PERFECT CUBES

Courtesy of Harold Hiken

Each of the following expressions is either a difference of perfect cubes or a sum of perfect cubes. Factor appropriately.

1) a³ + 1 11) 27a³ – 64b³

2) a³ – 1 12) 125t³ + 8s³

3) x³ – 8 13) 27r³ + 1000s³

4) m³ + 8 14) 216z³ – w³

5) p³ + q³ 15) 125m³ – 8p³

6) k³ – h³ 16) m⁶ – 8

7) y³ – 8x³ 17) 64y⁶ + 1

8) 8p³ + q³ 18) 8k⁶ – 27q³

9) 64p³ + n³ 19) 125z³ + 64r⁶

10) 27x³ – 1 20) (a – b)³ – (a + b)³

Answers:

1) (a + 1)(a² – a + 1)

2) (a – 1)(a² + a + 1)

3) (x – 2)(x² + 2x + 4)

4) (m + 2)(m² – 2m + 4)

5) (p + q)(p² – pq + q²)

6) (k – h)(k² + kh + h²)

7) (y – 2x)(y² + 2xy + 4x²)

8) (2p + q)(4p² – 2pq + q²)

9) (4p + n)(16p² – 4pn + n²)

10) (3x – 1)(9x² + 3x + 1)

11) (3a – 4b)(9a² + 12ab + 16b²)

12) (5t + 2s)(25t² – 10st + 4s²)

13) (3r + 10s)(9r² – 30rs + 100s²)

14) (6z – w)(36z² + 6zw + w²)

15) (5m – 2p)(25m² + 10mp + 4p²)

16) (m² – 2)(m⁴ + 2m² + 4)

17) (4y² + 1)(16y⁴ – 4y² + 1)

18) (2k² – 3q)(4k⁴ + 6k²q + 9q²)

19) (5z + 4r²)(25z² – 20r²z + 16r⁴)

20) ((a – b) – (a + b))((a – b)² + (a – b)(a + b) + (a + b)²) = -2b(3a² + b²)