Numsprite數靈精中文Play now
中文

Algebraic expressions · Ages 13–14

Factorising sums and differences of cubes worksheet

Part of: Factorisation

Learn the methods

In the print window, choose “Save as PDF” to download it. The answers print on the last page.

Factorising sums and differences of cubes worksheet

Algebraic expressions · Ages 13–14

Numsprite
numsprite.com

Name:Date:Score:／48

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

Part A: Working out and filling in (2 marks each)

  1. 1.x3 + 512 = (x + )(x2 − x + )

    Fill in each blank

    Answer:

  2. 2.x3 + 27 = (x + )(x2 − x + )

    Fill in each blank

    Answer:

  3. 3.8x3 − 125 = (x − )(x2 + x + )

    Fill in each blank

    Answer:

  4. 4.5x3 + 5 = (x + )(x2 − x + )

    Fill in each blank

    Answer:

  5. 5.64x3 − 125 = (x − )(x2 + x + )

    Fill in each blank

    Answer:

  6. 6.216 − x3 = (6 − x)( + 6x + x2)

    Answer:

Part B: Choose, or true or false (2 marks each)

  1. 7.x3 − 729 can be factorised as (x + 9)(x2 − 9x + 81).

    True or false?

    Answer:

  2. 8.Factorise x3 − 27. What do you get?

    • A. (x − 3)3
    • B. (x − 3)(x2 + 3x + 9)
    • C. (x − 3)(x2 + 6x + 9)
    • D. (x − 3)(x2 − 3x + 9)

    Choose the right answer

    Answer:

  3. 9.Factorise x3 − 64. What do you get?

    • A. (x − 4)(x2 + 4x + 16)
    • B. (x − 4)(x2 + 8x + 16)
    • C. (x − 4)(x2 − 4x + 16)
    • D. (x − 4)3

    Choose the right answer

    Answer:

  4. 10.2x3 − 54 factorised completely is 2(x − 3)(x2 + 3x + 9).

    True or false?

    Answer:

  5. 11.Factorise 125x3 − 8y3. What do you get?

    • A. (5x − 2y)(25x2 − 10xy + 4y2)
    • B. (5x − 2y)(5x2 + 10xy + 2y2)
    • C. (5x − 2y)3
    • D. (5x − 2y)(25x2 + 10xy + 4y2)

    Choose the right answer

    Answer:

  6. 12.6x3 − 750 factorised completely is 6(x − 5)(x2 + 10x + 25).

    True or false?

    Answer:

Part C: Word problems (4 marks each)

  1. 13.Tsz-hin factorises 5x3 + 40 as 5(x + 2)(x2 − 2x + 4). This is right.

    True or false?

    Working:

    Answer:

  2. 14.A closed hollow cube box is 18 cm along each outside edge and 12 cm inside. Factorising as a difference of cubes, the box takes cm3 of material.

    Working:

    Answer:

  3. 15.Which of these can't be factorised as a sum or difference of two cubes?

    • A. x3y3 + 216
    • B. x3 + 36
    • C. 64 − x3
    • D. 64x3 − y3

    Choose the right answer

    Working:

    Answer:

  4. 16.A wooden cube has edges x cm long, and a small cube with edges 3 cm long is cut from one corner. The volume left, (x3 − 27) cm3, can be written (x − )(x2 + x + ) cm3.

    Fill in each blank

    Working:

    Answer:

  5. 17.Factorise 6x3 − 384y3 completely: 6x3 − 384y3 = (x − y)(x2 + xy + y2).

    Fill in each blank

    Working:

    Answer:

  6. 18.Factorise 3x4 − 81x completely. Put the steps in order.

    • A. = 3x(x3 − 33)
    • B. = 3x(x − 3)(x2 + 3x + 9)
    • C. 3x4 − 81x = 3x(x3 − 27)

    Write the letters in the right order

    Working:

    Answer:

Show the answers and working

Factorising sums and differences of cubes worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 8, 8, 64

    512 = 83; with a = x and b = 8: x3 + 83 = (x + 8)(x2 − 8x + 64).

  2. 2. 3, 3, 9

    27 = 33; with a = x and b = 3: x3 + 33 = (x + 3)(x2 − 3x + 9).

  3. 3. 2, 5, 4, 10, 25

    8x3 = (2x)3 and 125 = 53. With a = 2x and b = 5: a2 = 4x2, ab = 10x, b2 = 25, so it is (2x − 5)(4x2 + 10x + 25).

  4. 4. 5, 1, 1, 1

    Take out the common factor 5 first: 5(x3 + 1); then the sum of cubes, 1 = 13: 5(x + 1)(x2 − x + 1).

  5. 5. 4, 5, 16, 20, 25

    64x3 = (4x)3 and 125 = 53. With a = 4x and b = 5: a2 = 16x2, ab = 20x, b2 = 25, so it is (4x − 5)(16x2 + 20x + 25).

  6. 6. 36

    With a = 6 and b = x: 216 − x3 = 63 − x3 = (6 − x)(36 + 6x + x2).

Part B: Choose, or true or false

  1. 7. False

    (x + 9)(x2 − 9x + 81) = x3 + 729, a sum of cubes; x3 − 729 factorises as (x − 9)(x2 + 9x + 81).

  2. 8. B. (x − 3)(x2 + 3x + 9)

    27 = 33: x3 − 33 = (x − 3)(x2 + 3x + 9).

  3. 9. A. (x − 4)(x2 + 4x + 16)

    64 = 43: x3 − 43 = (x − 4)(x2 + 4x + 16).

  4. 10. True

    2x3 − 54 = 2(x3 − 27) = 2(x − 3)(x2 + 3x + 9): the middle is ab = 3x, not 2ab.

  5. 11. D. (5x − 2y)(25x2 + 10xy + 4y2)

    125x3 = (5x)3 and 8y3 = (2y)3; with a = 5x and b = 2y it is (5x − 2y)(25x2 + 10xy + 4y2).

  6. 12. False

    6x3 − 750 = 6(x3 − 125) = 6(x − 5)(x2 + 5x + 25): the middle is ab = 5x, not 2ab.

Part C: Word problems

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

  1. 13. True

    5x3 + 40 = 5(x3 + 8); a sum of cubes' second bracket has a minus in the middle: 5(x + 2)(x2 − 2x + 4).

  2. 14. 4104

    The material's volume = 183 − 123 = (18 − 12)(182 + 18 × 12 + 122) = 6 × 684 = 4104.

  3. 15. B. x3 + 36

    36 = 62 is not a cube, so x3 + 36 is not a sum of two cubes.

  4. 16. 3, 3, 9

    27 = 33, so by the difference of cubes: x3 − 33 = (x − 3)(x2 + 3x + 9).

  5. 17. 6, 4, 4, 16

    Take out 6 first: 6(x3 − 64y3), and 64y3 = (4y)3; then the difference of cubes: 6(x − 4y)(x2 + 4xy + 16y2).

  6. 18. C → A → B

    Take out 3x first, write 27 as 33, then use the difference of cubes.

Numsprite · numsprite.com/en/worksheets/factorisation/factorising-sums-and-differences-of-cubes

Practise it in the gameFactorising sums and differences of cubes · Unlimited questions, no download
All worksheets for ages 13–14All worksheets