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GCSE · Algebra

4.8D Expanding and Factorising: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 65 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Expand 6(x + 4)

(Total for Question 1 is 1 mark)

Expand -5(3x - 2)

(Total for Question 2 is 1 mark)

Expand and simplify 2(x + 6) + 5(x - 2)

(Total for Question 3 is 2 marks)

Expand and simplify 7(x - 1) - 3(x + 5)

(Total for Question 4 is 2 marks)

Factorise fully 15x + 20

(Total for Question 5 is 1 mark)

Factorise fully 9x² - 15x

(Total for Question 6 is 2 marks)

Expand and simplify (x + 6)(x + 4)

(Total for Question 7 is 2 marks)

Expand and simplify (x - 5)(x + 9)

(Total for Question 8 is 2 marks)

Expand and simplify (x - 3)(x - 8)

(Total for Question 9 is 2 marks)

Expand and simplify (3x + 2)(x + 4)

(Total for Question 10 is 2 marks)

Expand and simplify (4x - 3)(2x + 5)

(Total for Question 11 is 2 marks)

Expand and simplify (x - 5)²

(Total for Question 12 is 2 marks)

Factorise x² + 9x + 14

(Total for Question 13 is 2 marks)

Factorise x² - 4x - 21

(Total for Question 14 is 2 marks)

Factorise x² + 3x - 40

(Total for Question 15 is 2 marks)

Factorise x² - 64

(Total for Question 16 is 2 marks)

Factorise fully 16x² - 81

(Total for Question 17 is 2 marks)

Factorise 6x² + 13x + 5

(Total for Question 18 is 3 marks)

Expand and simplify (x + 3)(x - 3)(x + 2)

(Total for Question 19 is 3 marks)

Nadia is solving the equation x² + 2x - 15 = 0 by factorising. Solve the equation, giving both solutions.

(Total for Question 20 is 3 marks)

Priya has designed a rectangular photo frame. The area of the front face of the frame is (x² + 5x - 24) cm², where x is in centimetres and x > 3.

Diagram NOT accurately drawn

(a)Factorise x² + 5x - 24 to write expressions, in terms of x, for the length and the width of the frame, given that the length is greater than the width.(2)

(b)Given that x = 7, work out the perimeter of the frame.(3)

(Total for Question 21 is 5 marks)

Prove algebraically that the difference between the squares of any two consecutive even numbers is always a multiple of 4.

(Total for Question 22 is 3 marks)

Simplify fully (x² + 3x - 10) / (x² - 4)

(Total for Question 23 is 3 marks)

Factorise fully x⁴ - 16

(Total for Question 24 is 3 marks)

Show that (2x + 3)(x - 4) - (2x - 1)(x + 5) simplifies to a linear expression. Factorise your answer fully.

(Total for Question 25 is 4 marks)

Mark scheme · 4.8D Expanding and Factorising: Fluency and Exam Drill

Question 1

B1 6x + 24 cao
Answer: 6x + 24

Question 2

B1 -15x + 10 oe cao
Answer: -15x + 10

Question 3

M1 2x + 12 + 5x - 10 (both brackets expanded correctly) oe
A1 7x + 2 cao
Answer: 7x + 2

Question 4

M1 7x - 7 - 3x - 15 (both brackets expanded, sign of second bracket correct) oe
A1 4x - 22 cao
Answer: 4x - 22

Question 5

B1 5(3x + 4) cao
Answer: 5(3x + 4)

Question 6

M1 correct HCF 3x identified oe
A1 3x(3x - 5) cao
Answer: 3x(3x - 5)

Question 7

M1 x² + 4x + 6x + 24, or at least 3 of the 4 terms correct, oe
A1 x² + 10x + 24 cao
Answer: x² + 10x + 24

Question 8

M1 x² + 9x - 5x - 45, or at least 3 of the 4 terms correct, oe
A1 x² + 4x - 45 cao
Answer: x² + 4x - 45

Question 9

M1 x² - 8x - 3x + 24, or at least 3 of the 4 terms correct, oe
A1 x² - 11x + 24 cao
Answer: x² - 11x + 24

Question 10

M1 3x² + 12x + 2x + 8, or at least 3 of the 4 terms correct, oe
A1 3x² + 14x + 8 cao
Answer: 3x² + 14x + 8

Question 11

M1 8x² + 20x - 6x - 15, or at least 3 of the 4 terms correct, oe
A1 8x² + 14x - 15 cao
Answer: 8x² + 14x - 15

Question 12

M1 writes as (x - 5)(x - 5) and attempts to expand, at least 3 of 4 terms correct
A1 x² - 10x + 25 cao
Answer: x² - 10x + 25

Question 13

M1 two numbers with sum 9 and product 14 identified (2 and 7) oe
A1 (x + 2)(x + 7) cao
Answer: (x + 2)(x + 7)

Question 14

M1 two numbers with sum -4 and product -21 identified (-7 and 3) oe
A1 (x - 7)(x + 3) cao
Answer: (x - 7)(x + 3)

Question 15

M1 two numbers with sum 3 and product -40 identified (8 and -5) oe
A1 (x + 8)(x - 5) cao
Answer: (x + 8)(x - 5)

Question 16

M1 recognises difference of two squares, √64 = 8 oe
A1 (x - 8)(x + 8) cao
Answer: (x - 8)(x + 8)

Question 17

M1 recognises difference of two squares, √16x² = 4x and √81 = 9 oe
A1 (4x - 9)(4x + 9) cao
Answer: (4x - 9)(4x + 9)

Question 18

M1 correct pair of factors of 6 and 5 tried (e.g. 2 and 3, 1 and 5) oe
M1 splits the middle term correctly, 6x² + 10x + 3x + 5 oe
A1 (2x + 1)(3x + 5) cao
Answer: (2x + 1)(3x + 5)

Question 19

M1 (x + 3)(x - 3) = x² - 9, or expands two of the three brackets correctly, oe
dM1 multiplies their quadratic by the remaining bracket, (x² - 9)(x + 2), with at least 3 correct terms oe
A1 x³ + 2x² - 9x - 18 cao
Answer: x³ + 2x² - 9x - 18

Question 20

M1 correctly factorises to (x + 5)(x - 3) oe
M1 sets each factor equal to zero, x + 5 = 0 or x - 3 = 0, oe
A1 x = -5 and x = 3 cao (both values)
Answer: x = -5 or x = 3

Question 21

(a) M1 two numbers with sum 5 and product -24 identified (8 and -3) oe
(a) A1 length = (x + 8) cm and width = (x - 3) cm oe
(a) Answer: length = (x + 8) cm, width = (x - 3) cm
(b) M1 substitutes x = 7 into both expressions to find length = 15 cm and width = 4 cm oe
(b) M1 uses perimeter = 2(length + width) oe
(b) A1 38 cm cao
(b) Answer: 38 cm

Question 22

M1 lets the two consecutive even numbers be 2n and 2n + 2 (or equivalent) oe
M1 expands and simplifies (2n + 2)² - (2n)² to reach 8n + 4 oe
C1 factorises to 4(2n + 1) and concludes this is a multiple of 4 for all integer n, cso
Answer: (2n + 2)² - (2n)² = 4(2n + 1), which is a multiple of 4 for every integer n

Question 23

M1 factorises the numerator to (x + 5)(x - 2) oe
M1 factorises the denominator to (x - 2)(x + 2) oe
A1 (x + 5)/(x + 2) cao, with (x - 2) cancelled
Answer: (x + 5)/(x + 2)

Question 24

M1 recognises difference of two squares, x⁴ - 16 = (x² - 4)(x² + 4) oe
dM1 factorises x² - 4 further as (x - 2)(x + 2) oe
A1 (x - 2)(x + 2)(x² + 4) cao, fully factorised (x² + 4 does not factorise further)
Answer: (x - 2)(x + 2)(x² + 4)

Question 25

M1 expands (2x + 3)(x - 4) correctly to 2x² - 5x - 12 oe
M1 expands (2x - 1)(x + 5) correctly to 2x² + 9x - 5 oe
dM1 subtracts correctly, taking care with signs, to reach -14x - 7, cso
A1 -7(2x + 1) cao, fully factorised
Answer: -7(2x + 1)

Expanding and Factorising: Fluency and Exam Drill - Worksheets, Questions and Revision