Simplify and Factorise – Exam Answers

1.	(a)	(i)	a2 – 2ab + b2	5
B1 for a2 – 2ab + b2
(ii)	Printed result
	LHS = 
 = 
M1 for 
A1 for a convincing completion to printed result
(iii)	20 012.5
	a = 102.5, b = 97.5
a + b = 200 , a – b = 5
Require 
M1 for either a + b = 200 or a − b = 5
A1 for 20 012.5
[5]

2.	(i)	m9	5
B1 for 
(ii)	p4
B1 for 
(iii)	10x4y9
B2 for 10x4y9
(iv)	4(k + 8)
B1 for 4(k + 8)
[5]

3.		2
	
M1 for at least 3 of the 4 terms correct
A1 cao
	25	2
M1 for recognising 3.47 + 1.53 (= 5)
A1 cao
[4]

4.	(a)	-6x + 23	2
	6x + 8 -12x + 15
M1 for 3 of the 4 terms 6x, +8, -12x, +15 correct
A1 cao
(b)	32x5y15	2
B2 cao
(B1 for two of 32, x5, y15)
(c)		3
	
M1 for k(n + 1) (n – 1)
M1 dep for  = n - 1
A1 for 
[7]

5.	(a)	(3x – 1)2	2
B1 for (3x – 1)(..x….) cao
B2 for (3x - 1)2 cao
(b)		3
	
B1 for correct factorisation of numerator
M1 for cancelling of common factors
A1 cao
[5]

6.	(a)	x(x – 3)	2
B2 for x(x – 3)
(B1 for x (…….))
(b)	k3.	1
B1 for k3.
(c)	(i)	7x – 1	4
	4x + 20 + 3x – 21
M1 for three of 4 terms 4x + 20 + 3x – 21 (or better)
A1 for 7x – 1
(ii)	x2 + 5xy+ 6y2
	x2 + 3xy+ 2xy + 6y2
M1 for three of 4 terms x2 + 3yx+ 2xy + 6y2
A1 for x2 + 5xy 6y2
(d)	(p + q)(p + q +5)	1
B1 for (p + q)(p + q +5)
[8]

7.	(a)	k3.	1
B1 for k3.
(b)	(i)	7x – 1	4
	4x + 20 + 3x – 21
M1 for three of 4 terms 4x + 20 + 3x – 21 (or better)
A1 for 7x – 1
(ii)	x2 + 5xy+ 6y2
	x2 + 3xy+ 2xy + 6y2
M1 for three of 4 terms x2 + 3xy+ 2xy + 6y2
A1 for x2 + 5xy+ 6y2
(c)	(p + q)(p + q +5)	1
B1 for (p + q)(p + q +5)
(d)	m8	1
B1 for m8.
(e)	6r3t6	2
B2 for 6r3t6
(B1 for .....r3t6 or for 6...t6)
[9]

8.	(a)	7	3
	40 – x = 3(4 + x)
40 – x = 12 + 3x
40 – 12 = x + 3x
4x = 28
M1 multiplying through by 3:

A1 40 – 12= x + 3x
A1 cao
(b)		3
	 = 
B1 for (2x – 3)(2x + 3)
B1 for 2x(2x – 3) or (2x + 0)(2x + 3)
B1 cao
[6]

9.	(x + 1)(x + 6)	2
M1 (x + a)(x + b) with ab = 6
A1 cao
[2]

10.	(a)	a7	1
B1 accept a4 + 3
(b)	15x3y4	2
B2 cao
(B1 for two of 15, x3, y4 in a product)
(c)	x – 1	1
B1 cao
(d)	(a + 3b)(a – 3b)	2
B2 for (a + 3b)(a – 3b)
(B1 for (a ± 3b)(a ± 3b)
[6]

11.	(a)	81x4y8	2
	34x4y8
B2 for 81x4y8
(B1 for 2 of 81, x4, y8)

(b)		3
	
B1 for x(x – 3)
B1 for (x – 5)(x – 3)
B1 cao
[5]

12.	
		3
B1 for (5 + x)(5 – x)
B1 for 5(5 + x)
B1 cao for oe
[3]

13.	(a)	(x + 2)(x + 4)	2
M1 (x ± 2)(x ± 4)
A1 cao
(b)	–2, –4	1
B1ft from (a) or –2, –4
[3]

14.	(a)	(i)	x9	1
B1 cao
(ii)	p5	1
B1 cao
(iii)	12 s6 t5	2
B2 cao
(B1 for two of 12, s6, t5 in a product)
(iv)	q12	1
B1 cao
(b)	6g – 3	1
B1 cao
(c)	2d2 + 6d	2
B2 cao (B1 for 2d2 or 6d)

(d)	x2 +3x + 2x + 6	2
x2 + 5x + 6
B2 for x2 + 5x + 6
(B1 for 3 out of 4 parts correct in working)
[10]

15.	(a)	27x6y12
27x6y12	2
B2 for fully correct
B1 for 2 of 27, x6, y12 correct in a 3 term product
(b)	6x2 + 15x – 4x –10
6x2 + 11x – 10	2
B2 for fully correct
(B1 for 3 out of 4 terms correct in working including signs or	4 terms correct, incorrect signs)
(c)	
	2
B2 for 
(B1 for x(x + 2) or (x + 2)(x + 3) seen)
[6]

16.	(x – 5)(x + 3)	2
B1 (x ± 5)(x ± 3)
B1 cao
[2

17.	(a)	(x - 5)(x + 3)	2
B1 (x ± 5)(x ± 3)
B1 cao
(b)	5 or –3	1
B1 cao
[3]

18.		3
	
B1 numerator
B1 indep denominator
B1 cao
[3]

19.	(i)	m9	4
B1 for m9
(ii)	p4
B1 for p4
(iii)	q5
B1 for q5
(iv)	4(k + 8)
B1 for 4(k + 8)
[4]

20.	(a)	4y–2 OR 	2
B2 cao
(B1 for 4yn, where n is an integer OR ay-2)
(b)	6w7x3	2
B2 cao
(B1 for any 2 correct of 6, w7, x3)
[4]

21.	(a)	x2 – 2x – 24	2
	x2 – 6x + 4x – 24
M1 for any 3 terms correct
A1 cao
(b)	6x (2x – 3y)	2
B2 for 6x(2x – 3y)
(B1 for either x(12x – 18y) OR
2x(6x - 9y) OR 3x(4x - 6y)
OR 6x(ax - by) where either a ¹ 2 or b ¹ 3
[4]

22.		3
	
B3 for 
[B1 for x(2x + 3) seen
AND B1 for (2x - 3)(2x + 3) seen]
[3]

23.	3(x + 2) oe	1
B1 for 3(x + 2) or better
[1]

24.	35x4y6	2
B2
(B1 for any 2 of 35, x4, y6)
[2]

25.	(a)	x(x + 1)	1
B1 [accept (x + 0)(x + 1)]
(b)	(y + 5)(y – 7)	2
B2 for (y + 5)(y – 7)
[B1 for (y ± 5)(y ± 7)]
[3]

26.	5ab4	2
B2 cao
(B1 for 2 of 5, a, b4 correct)
[2]

27.		2
	
M1 for 
A1 cao
[2]

28.	(a)	k3	1
B1 for k3
(b)	7x − 1	2
	4x + 20 + 3x − 21
M1 for three of 4 terms 4x + 20 + 3x − 21 (or better)
A1 for 7x − 1
[3]

29.	(a)	12x2 + 11x + 2	2
	12x2 + 8x + 3x + 2
M1 for expansion (condone one error)
A1 cao
(b)	3x(x + 2y)	2
B2 cao
(B1 for 3x(	) or __(x + 2y) or 3(x2 + 2xy) or x(3x + 6y))
[4]

30.	(a)	(x – 5) (2x – 7)	2
	(x – 5) (2x – 10 + 3)
M1 for (x – 5) (2(x – 5) + 3) or for identifying (x – 5) as a common factor or 2x2 – 17x + 35
A1 cao
(b)		1
B1 cao
[3]

31.	81x4y8	2
	34x4y8
B2 for 81x4y8
(B1 for 2 of 81, x4, y8)
[2]

32.	(a)	p6	1
B1 cao
(b)	w12	1
B1 cao
(c)	5e5f3	2
B2 cao
(B1 for either e5 or f3)
[4]

33.	(a)		3
B1 for x-5 or 
(b)	
B2 for w8y6
(B1 for any 2 of 12, w8, y6 correct)
[3]

34.	
	2
M1 for (2x + 1)(2x – 1)
A1 cao
[2]

35.	(i)	12 s6 t5
B2 cao
(B1 for two of 12, s6, t5 in a product)
(ii)	q12
B1 cao
[2]

36.	(a)	(i)	x9	1
B1 cao
(ii)	p5	1
B1 cao
(iii)	12 s6 t5	2
B2 cao
(B1 for two of 12, s6, t5 in a product)
(iv)	q12	1
B1 cao
(b)	2d2 + 6d	2
B2 cao
(B1 for 2d2 or 6d)
[7]

37.	6x2 – 4x + 15x – 10
6x2 + 11x –10	2
B2 for fully correct
(B1 for 3 out of 4 terms correct in working, including signs
or 4 terms correct, incorrect signs)
[2]