EDEXCEL GCSE MATHS · HIGHER TIER
Exam Practice: More Expanding and Factorising
More expanding and factorising · Algebra · Lesson 7 of 7 · 16 marks · 30 minutes
Name Date
Answer all questions. Show your working. All questions are non-calculator.
Question 1 NON-CALCULATOR [2 marks]
Factorise 3x² + 10x + 8
Question 2 NON-CALCULATOR [3 marks]
Expand and simplify (x + 4)(x − 1)(x + 2)
Question 3 NON-CALCULATOR [2 marks]
Factorise fully 50 − 2x²
Question 4 NON-CALCULATOR [3 marks]
Simplify fully (x² + 3x − 10)/(x² − 4)
Question 5 NON-CALCULATOR · PROVE [3 marks]
Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for all positive integer values of n.
Question 6 NON-CALCULATOR · SHOW THAT [3 marks]
The diagram shows a square of side (2x + 3) cm with a square of side (x + 1) cm cut from one corner. Show that the shaded area, in cm², can be written as (3x + 4)(x + 2).
Answers
Check your answer only once you have written one.
Question 1 [2 marks]
ac = 24; 6 and 4 multiply to 24 and add to 10. 3x² + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2)
▸ (3x ± 4)(x ± 2), or a correct split of the middle term M1
▸ (3x + 4)(x + 2) A1
Question 2 [3 marks]
(x + 4)(x − 1) = x² + 3x − 4. (x² + 3x − 4)(x + 2) = x³ + 2x² + 3x² + 6x − 4x − 8 = x³ + 5x² + 2x − 8
▸ Two brackets expanded correctly M1
▸ At least 6 correct terms from multiplying by the third bracket M1
▸ x³ + 5x² + 2x − 8 A1
Question 3 [2 marks]
2(25 − x²) = 2(5 + x)(5 − x)
▸ 2(25 − x²), or (10 + 2x)(5 − x) M1
▸ 2(5 + x)(5 − x) A1
Question 4 [3 marks]
x² + 3x − 10 = (x + 5)(x − 2) and x² − 4 = (x + 2)(x − 2). Cancelling (x − 2): (x + 5)/(x + 2).
▸ (x + 5)(x − 2) M1
▸ (x + 2)(x − 2) M1
▸ (x + 5)/(x + 2) A1
Question 5 [3 marks]
(2n + 1)² = 4n² + 4n + 1 and (2n − 1)² = 4n² − 4n + 1. Subtracting: 8n. 8n = 8 × n, so it is always a multiple of 8.
▸ One bracket expanded correctly M1
▸ 8n A1
▸ A concluding statement: 8n is a multiple of 8 C1
Question 6 [3 marks]
Shaded area = (2x + 3)² − (x + 1)². This is a difference of two squares: ((2x + 3) + (x + 1))((2x + 3) − (x + 1)) = (3x + 4)(x + 2).
▸ (2x + 3)² − (x + 1)² M1
▸ Uses the difference of two squares, or expands both to reach 3x² + 10x + 8 M1
▸ Correctly reaches (3x + 4)(x + 2) A1