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Algebraic expressions · Ages 13–14

Factorisation worksheet

Covers: Taking out a common factor, Factorising a difference of two squares, Factorising perfect squares, Factorising by grouping, The cross method: leading coefficient 1, The cross method: other leading coefficients, Factorising sums and differences of cubes

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Factorisation worksheet

Algebraic expressions · Ages 13–14

Numsprite
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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. 1.32x2 + 4x = 4x(x + )

    Fill in each blank

    Answer:

  2. 2.7x2 − 63 = (x + )(x − )

    Fill in each blank

    Answer:

  3. 3.6x2 − 108x + 486 = (x − )2

    Fill in each blank

    Answer:

  4. 4.xy + 6x + 8y + 48 = (x + )(y + )

    Fill in each blank

    Answer:

  5. 5.x2 + 15xy + 54y2 = (x + 6y)(x + y)

    Answer:

  6. 6.Factorising 2x2 + 17x + 8 by the cross method, with 2x and x on the left and 1 and 8 on the right: across, 2x × 8 = x and x × 1 = x, which add to x.

    Fill in each blank

    Answer:

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

  1. 7.Factorise 54x2 − 30x. What do you get?

    • A. 6x(9x − 5)
    • B. 6(9x2 − 5x)
    • C. 6x(9x − 30)
    • D. x(54x − 30)

    Choose the right answer

    Answer:

  2. 8.Factorise 25x2 − 1. What do you get?

    • A. (5x + 1)(5x − 1)
    • B. (25x + 1)(25x − 1)
    • C. (5x + 1)2
    • D. (5x − 1)2

    Choose the right answer

    Answer:

  3. 9.Factorise 9x2 − 30xy + 25y2. What do you get?

    • A. (3x − 5y)2
    • B. (3x + 5y)(3x − 5y)
    • C. (5x − 3y)2
    • D. (3x + 5y)2

    Choose the right answer

    Answer:

  4. 10.xy + 9x + 3y + 27 can be factorised as (x + 9)(y + 3).

    True or false?

    Answer:

  5. 11.x2 − x − 20 = (x + 5)(x − 4)

    True or false?

    Answer:

  6. 12.Factorise x3 − 216. What do you get?

    • A. (x − 6)3
    • B. (x − 6)(x2 − 6x + 36)
    • C. (x − 6)(x2 + 6x + 36)
    • D. (x − 6)(x2 + 12x + 36)

    Choose the right answer

    Answer:

Part C: Word problems (4 marks each)

  1. 13.A square lawn has sides x m long, and a square pond 5 m on a side is dug in one corner. The area left, (x2 − 25) m2, can be written (x + )(x − ) m2.

    Fill in each blank

    Working:

    Answer:

  2. 14.A square rug has an area of (x2 + 4x + 4) m2. Its side is (x + ) m long; when x = 11, the side is m.

    Fill in each blank

    Working:

    Answer:

  3. 15.Group in pairs to work it out: 39 × 85 + 39 × 15 + 40 × 85 + 40 × 15 =

    Working:

    Answer:

  4. 16.Factorise x2 + 13x + 42. Put the steps in order.

    • A. x2 + 13x + 42 = (x + 6)(x + 7)
    • B. Find two numbers: product = 42, sum = 13
    • C. Check: (x + 6)(x + 7) = x2 + 7x + 6x + 42 = x2 + 13x + 42
    • D. 6 × 7 = 42, 6 + 7 = 13

    Write the letters in the right order

    Working:

    Answer:

  5. 17.(x + 1) is a factor of 5x2 + 8x + 3.

    True or false?

    Working:

    Answer:

  6. 18.May factorises 4x3 + 256 as 4(x + 4)(x2 + 4x + 16). This is right.

    True or false?

    Working:

    Answer:

Show the answers and working

Factorisation worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 8, 1

    32x2 ÷ 4x = 8x and 4x ÷ 4x = 1: taking out a whole term leaves 1 behind.

  2. 2. 7, 3, 3

    Take out the common factor 7 first: 7(x2 − 9); then the difference of squares: 7(x + 3)(x − 3).

  3. 3. 6, 9

    Take out the common factor 6: 6(x2 − 18x + 81); inside is a perfect square: 6(x − 9)2.

  4. 4. 8, 6

    In pairs: x(y + 6) + 8(y + 6) = (x + 8)(y + 6).

  5. 5. 9

    Find the two numbers as for x2 + 15x + 54: 6 and 9, each with a y.

  6. 6. 16, 1, 17

    16x + x = 17x, the middle term itself, so 2x2 + 17x + 8 = (2x + 1)(x + 8).

Part B: Choose, or true or false

  1. 7. A. 6x(9x − 5)

    The H.C.F. of 54 and 30 is 6, and both terms have x: the common factor is 6x. 54x2 ÷ 6x = 9x and 30x ÷ 6x = 5.

  2. 8. A. (5x + 1)(5x − 1)

    25x2 − 1 = (5x)2 − 12 = (5x + 1)(5x − 1).

  3. 9. A. (3x − 5y)2

    9x2 = (3x)2, 25y2 = (5y)2 and the middle 30xy = 2 × 3x × 5y, with a minus: (3x − 5y)2.

  4. 10. False

    xy + 9x + 3y + 27 = x(y + 9) + 3(y + 9) = (x + 3)(y + 9).

  5. 11. False

    The two numbers with product −20 and sum −1 are 4 and −5, so it is (x + 4)(x − 5).

  6. 12. C. (x − 6)(x2 + 6x + 36)

    216 = 63: x3 − 63 = (x − 6)(x2 + 6x + 36).

Part C: Word problems

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

  1. 13. 5, 5

    x2 − 25 = x2 − 52 = (x + 5)(x − 5).

  2. 14. 2, 13

    x2 + 4x + 4 = (x + 2)2, so the side is x + 2; when x = 11, 11 + 2 = 13.

  3. 15. 7900

    = 39(85 + 15) + 40(85 + 15) = (39 + 40) × 100 = 79 × 100 = 7900.

  4. 16. B → D → A → C

    Say what is needed, find the two numbers, write the two factors, then expand to check.

  5. 17. True

    5x2 + 8x + 3 = (5x + 3)(x + 1): its linear factors are 5x + 3 and x + 1.

  6. 18. False

    4x3 + 256 = 4(x3 + 64); a sum of cubes' second bracket has a minus in the middle: 4(x + 4)(x2 − 4x + 16).

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