FACTORING SUM AND DIFFERENCE OF CUBES WORKSHEET

Factor each completely. 

1)  m³ + 8

2)  a³ - 125

3)  x³ + 8y³

4)  8x³ - 125y³

5)  27x³ + 64y³

6)  2m³ - 54n³

7)  256p³ + 500q³

8)  648a³ - 3b³

Find the following products:

9) (3x + 2y)(9x² – 6xy + 4y²)

10) (4x – 5y)(16x² + 20xy + 25y²)

11) (7p⁴ + q)(49p⁸ – 7p⁴q + q²)

12) (x/2 + 2y)(x²/4 – xy + 4y²)

13) (3/x – 5/y)(9/x² + 25/y² + 15/xy)

14) (2/x + 3x)(4/x² + 9x² – 6)

15) (3/x – 2x²)(9/x² + 4x⁴ – 6x)

16)  (1 – x)(1 + x + x²)

17)  If a + b = 10 and ab = 16, find the value of a² – ab + b² and a² + ab + b².

18)  If a + b = 8 and ab = 6, find the value of a³ + b³.

Detailed Answer Key

1. Answer :

m³ + 8  =  m³ + 2³

=  m³ + 2³

=  (m + 2)(m² - 2m + 2²)

=  (m + 2)(m² - 2m + 4)

2. Answer :

a³ - 125  =  a³ - 5³

=  a³ - 5³

=  (a - 5)(a² + 5a + 5²)

=  (a - 5)(a² + 5a + 25)

x³ + 8y³

3. Answer :

x³ + 8y³  =  x³ + 2³y³

=  x³ + (2y)³

=  (x + 2y)[x² - (x)(2y) + (2y)²]

=  (x + 2y)(x² - 2xy + 4y²)

8x³ - 125y³

4. Answer :

8x³ - 125y³  =  2³x³ - 5³y³

=  (2x)³ - (5y)³

=  (2x - 5y)[(2x)² + (2x)(5y) + (5y)²]

=  (2x - 5y)(4x² + 10xy + 25y²)

5. Answer :

27x³ + 64y³  =  3³x³ + 4³y³

=  (3x)³ + (4y)³

=  (3x + 4y)[(3x)² - (3x)(4y) + (4y)²]

=  (3x + 4y)(9x² - 12xy + 16y²)

6. Answer :

2m³ - 54n³  =  2(m³ - 27n³)

=  2(m³ - 3³n³)

=  2[m³ - (3n)³]

=  2(m - 3n)[m² + (m)(3n) + (3n)²]

=  2(m - 3n)(m² + 3mn + 9n²)

7. Answer :

256p³ + 500q³  =  4(64p³ + 125q³)

=  4[(4p)³ + (5q)³]

=  4(4p + 5q)[(4p)² - (4p)(5q) + (5q)²]

=  4(4p + 5q)(16p² - 20pq + 25q²)

8. Answer :

648a³ - 3b³  =  3(216a³ - b³)

=  3[(6a)³ - b³]

=  3(6a - b)[(6a)² + (6a)(b) + b²]

=  4(6a - b)(36a² + 6ab + b²)

9. Answer :

= (3x + 2y)(9x² – 6xy + 4y²)

= (3x + 2y)(3²x² – 3x(2y) + 2²y²)

= (3x + 2y)((3x)² – 3x(2y) + (2y)²) 

Looks like (a + b) (a² - ab + b²), then we write it as a³ + b³

= (3x)³ + (2y)³

= 27x³ + 8y³

10. Answer :

= (4x – 5y)(16x² + 20xy + 25y²)

= (4x - 5y)(4²x² – 4x(5y) + 5²y²)

= (4x - 5y)((4x)² – 4x(5y) + (5y)²) 

Looks like (a - b) (a² + ab + b²), then we write it as a³ - b³

= (4x)³ - (5y)³

= 64x³ - 125y³

11. Answer :

= (7p⁴ + q)(49p⁸ – 7p⁴q + q²)

= (7p⁴ + q) [7²(p⁴)² – 7p⁴(q) + q²]

= (7p⁴ + q) ((7p⁴)² – 7p⁴(q) + q²]

Looks like (a + b) (a² - ab + b²), then we write it as a³ + b³

= (7p⁴)³ - q³

= 343p¹² - q³

12. Answer :

=  (x/2 + 2y)(x²/4 – xy + 4y²)

=  (x/2 + 2y)[ (x/2)² – (x/2)(2y) + (2y)² ]

Looks like (a + b) (a² - ab + b²), then we write it as a³ + b³

= (x/2)³ - (2y)³

= x³/8 - 8y³

13. Answer :

=  (3/x – 5/y)(9/x² + 25/y² + 15/xy)

= (3/x – 5/y)[(3/x)² + (5/y)² + (3/x)(5/y)]

Looks like (a - b) (a² + ab + b²), then we write it as a³ - b³

= (3/x)³ - (5/y)³

= 27/x³ - 125/y³

14. Answer :

= (2/x + 3x)(4/x² + 9x² – 6)

= (2/x + 3x)[(2/x)² + (3x)² - (2/x)(3x)]

Looks like (a + b) (a² - ab + b²), then we write it as a³ + b³

= (2/x)³ + (3x)³

= 8/x³ + 27x³

15. Answer :

= (3/x – 2x²)(9/x² + 4x⁴ + 6x)

= (3/x – 2x²)[(3²/x² + 2²(x²)² + 6x)

= (3/x – 2x²)[(3/x)² + (2x²)² + (3/x)2x²)

Looks like (a - b) (a² + ab + b²), then we write it as a³ - b³

= (3/x)³ - (2x²)³

= 27/x³ - 8x⁶

16. Answer :

= (1 – x)(1 + x + x²)

= (1 – x)(1 + x(1) + x²)

= 1 - x³

17 Answer :

If a + b = 10 and ab = 16, find the values of a² – ab + b², a³ + b³ and a² + ab + b².

a + b = 10 -----(1)

ab = 16  -----(2)

Formula for

a³ + b³ = (a + b)(a² - ab + b²) ----(1)

Here solving for a² + b² = (a + b)²- 2ab

a² + b² = 10²- 2(16)

= 100 - 32

a² + b² = 68

Applying these values in (1), we get

= (10)(68 - 16)

= 10(52)

a³ + b³ = 520

a² - ab + b² = 68 - 16

= 52

a² + ab + b² = 68 + 16

= 84

18 Answer :

If a + b = 8 and ab = 6, find the value of a³ + b³.

a + b = 8 -----(1)

ab = 6  -----(2)

Formula for

a³ + b³ = (a + b)(a² - ab + b²)

Here solving for a² + b² = (a + b)²- 2ab

a² + b² = 8²- 2(6)

= 64 - 12

= 52

= (8)(52 - 6)

= 8(46)

= 368

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