Algebraic expressions · Ages 13–14

The sum and difference of two cubes worksheet

Part of: Polynomials & identities

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Algebraic expressions · Ages 13–14

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.(x + 3)(x² − 3x + 9) = x³ +
Answer:
2.27x³ − 8y³ = (x)³ − (y)³
Fill in each blank
Answer:
3.(2x + 5)(4x² − 10x + 25) = x³ +
Fill in each blank
Answer:
4.Work it out quickly with the difference of two cubes: 5³ − 2³ = (5 − 2) ×
Answer:
5.(2x + 9)(4x² − 18x + 81) = x³ +
Fill in each blank
Answer:
6.(x + 3)³ − (x³ + 27) = x² + x
Fill in each blank
Answer:
7.(x + 9)³ − (x³ + 729) = x² + x
Fill in each blank
Answer:

13.A cube with edges x cm long and a cube with edges 3 cm long have a total volume of (x³ + 27) cm³ = (x + 3)(x² − 3x + 9) cm³. When x = 10, the second factor is and the total volume is cm³.
Fill in each blank
Working:
Answer:
14.A solid wooden cube with edges 11 cm long has a cube with edges 7 cm long cut out of it. Which of these gives the volume left, in cm³?
- A. 4³
- B. 4 × 247
- C. 4 × 93
- D. 4 × 170
Choose the right answer
Working:
Answer:
15.Prove (x − 3)(x² + 3x + 9) ≡ x³ − 27. Put the steps in order.
- A. = x³ − 27
- B. Left side = x(x² + 3x + 9) − 3(x² + 3x + 9)
- C. = right side
- D. = x³ + 3x² + 9x − 3x² − 9x − 27
Write the letters in the right order
Working:
Answer:
16.Chi-keung says: (x + 6)(x² − 6x + 36) = x³ + 216.
True or false?
Working:
Answer:
17.Two solid metal cubes have edges 13 cm and 7 cm long. Using a³ − b³ ≡ (a − b)(a² + ab + b²), their volumes differ by cm³.
Working:
Answer:
18.Two solid metal cubes have edges 10 cm and 8 cm long. Using a³ − b³ ≡ (a − b)(a² + ab + b²), their volumes differ by cm³.
Working:
Answer:

Show the answers and working

The sum and difference of two cubes worksheet: Answers and working

1. 27
It is a sum of cubes with a = x and b = 3: x³ + 3³ = x³ + 27. Multiplying term by term gives the same: −3x² and +3x², +9x and −9x cancel.
2. 3, 2
27 = 3 × 3 × 3 = 3³ and 8 = 2³, so 27x³ = (3x)³ and 8y³ = (2y)³.
3. 8, 125
With a = 2x and b = 5, the second bracket is exactly a² − ab + b², so the product is (2x)³ + 5³ = 8x³ + 125.
4. 39
a³ − b³ ≡ (a − b)(a² + ab + b²): the second factor is 5² + 5 × 2 + 2² = 39.
5. 8, 729
With a = 2x and b = 9, the second bracket is exactly a² − ab + b², so the product is (2x)³ + 9³ = 8x³ + 729.
6. 9, 27
(x + 3)³ = x³ + 9x² + 27x + 27, two middle terms more than x³ + 27: a sum of cubes is not the cube of a sum.
7. 27, 243
(x + 9)³ = x³ + 27x² + 243x + 729, two middle terms more than x³ + 729: a sum of cubes is not the cube of a sum.

8. D. x³ − 125
The difference of cubes: (x − 5)(x² + 5x + 25) = x³ − 5³ = x³ − 125.
9. True
a³ + b³ ≡ (a + b)(a² − ab + b²): the second bracket's middle term is −2x, the opposite sign to the first bracket's.
10. False
a³ + b³ ≡ (a + b)(a² − ab + b²): the second bracket's middle term is −3x, the opposite sign to the first bracket's.
11. False
9³ + 8³ = (9 + 8)(9² − 9 × 8 + 8²), so it is always divisible by 17; and 9³ + 8³ = 1241.
12. False
15³ + 7³ = (15 + 7)(15² − 15 × 7 + 7²), so it is always divisible by 22; and 15³ + 7³ = 3718.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 79, 1027
10² − 3 × 10 + 9 = 79; the total volume = (10 + 3) × 79 = 1027, the same as 10³ + 27.
14. B. 4 × 247
The volume left = 11³ − 7³ = (11 − 7)(11² + 11 × 7 + 7²) = 4 × 247 = 988.
15. B → D → A → C
Split it with the distributive law, then multiply term by term; the 3x² pair and the 9x pair cancel, leaving x³ − 27, the right side.
16. True
The second bracket's middle term takes the opposite sign to the first bracket's: (x + 6)(x² − 6x + 36) = x³ + 216 and (x − 6)(x² + 6x + 36) = x³ − 216.
17. 1854
13³ − 7³ = (13 − 7)(13² + 13 × 7 + 7²) = 6 × 309 = 1854.
18. 488
10³ − 8³ = (10 − 8)(10² + 10 × 8 + 8²) = 2 × 244 = 488.

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