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6.5D Expanding Triple Brackets: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Expand and simplify (x - 3)(x + 6).

(Total for Question 1 is 2 marks)

Expand and simplify (x + 1)(x + 3)(x + 4), giving your answer in the form x³ + ax² + bx + c.

(Total for Question 2 is 3 marks)

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

(Total for Question 3 is 3 marks)

Expand and simplify (x - 1)(x - 2)(x - 3).

(Total for Question 4 is 3 marks)

Expand and simplify (2x + 1)(x + 3)(x + 4).

(Total for Question 5 is 4 marks)

Show that (x - 3)(x + 4)(x + 1) = x³ + 2x² - 11x - 12.

(Total for Question 6 is 3 marks)

A cuboid has length (x + 2) cm, width (x + 3) cm and height (x + 5) cm.

(a)Show that the volume, V cm³, of the cuboid is given by V = x³ + 10x² + 31x + 30.(3)

(b)Work out the volume of the cuboid when x = 1.(2)

(Total for Question 7 is 5 marks)

Given that (x + 4)(x - 1)(x + 2) = x³ + 5x² + kx - 8, find the value of k.

(Total for Question 8 is 2 marks)

Expand and simplify (x + 2)³, giving your answer in the form x³ + ax² + bx + c.

(Total for Question 9 is 4 marks)

Expand and simplify (x + 5)(x - 5)(x + 6). Give your answer in the form x³ + ax² + bx + c.

(Total for Question 10 is 4 marks)

Expand and simplify (x + 4)³, giving your answer in the form x³ + ax² + bx + c.

(Total for Question 11 is 4 marks)

f(x) = (x - 2)(x + 5)(x - 3)

(a)Show that f(x) = x³ - 19x + 30.(3)

(b)Hence write down the coefficient of x² in the expansion of f(x).(1)

(Total for Question 12 is 4 marks)

Given that (x + a)(x + 4)(x - 3) = x³ + 3x² - 10x - 24, find the value of a.

(Total for Question 13 is 4 marks)

Expand and simplify (3 - x)(x + 1)(x + 4).

(Total for Question 14 is 4 marks)

Expand and simplify (x + 2)(3x - 1)(x - 4).

(Total for Question 15 is 5 marks)

f(x) = (x - 5)(x + 2)(x + 1)

(a)Expand f(x) fully, giving your answer in the form x³ + ax² + bx + c.(3)

(b)Find f(0).(2)

(c)Hence write down the coordinates of the point where the graph of y = f(x) crosses the y-axis.(1)

(Total for Question 16 is 6 marks)

Expand and simplify (3x + 1)(x - 2)(2x + 3).

(Total for Question 17 is 5 marks)

Show that (x + 2)(x + 3)(x + 4) - (x - 2)(x - 3)(x - 4) = 18x² + 48.

(Total for Question 18 is 4 marks)

A cuboid has dimensions (x + 4) cm, (x - 1) cm and (x + 3) cm, where x > 1.

(a)Show that the volume, V cm³, of the cuboid is given by V = x³ + 6x² + 5x - 12.(3)

(b)The volume of the cuboid is 30 cm^3. Show that x³ + 6x² + 5x - 42 = 0.(1)

(c)Show that x = 2 satisfies the equation in part (b), and hence write down the three dimensions of the cuboid.(2)

(Total for Question 19 is 6 marks)

Prove that n(n + 1)(n + 2) is a multiple of 6 for every positive integer n.

(Total for Question 20 is 3 marks)

Mark scheme · 6.5D Expanding Triple Brackets: Fluency and Exam Drill

Question 1

M1 attempt to multiply out the two brackets with at least 3 of the 4 terms correct, e.g. x² + 6x - 3x (- 18) oe
A1 x² + 3x - 18 cao
Answer: x² + 3x - 18

Question 2

M1 expands two of the three brackets correctly, e.g. (x+3)(x+4) = x²+7x+12 oe
M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
A1 x³ + 8x² + 19x + 12 cao
Answer: x³ + 8x² + 19x + 12

Question 3

M1 expands two of the three brackets correctly, e.g. (x-4)(x+5) = x²+x-20 oe
M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
A1 x³ + 3x² - 18x - 40 cao
Answer: x³ + 3x² - 18x - 40

Question 4

M1 expands two of the three brackets correctly, e.g. (x-1)(x-2) = x²-3x+2 oe
M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
A1 x³ - 6x² + 11x - 6 cao
Answer: x³ - 6x² + 11x - 6

Question 5

M1 expands (x+3)(x+4) correctly to get x²+7x+12 oe
M1 (dep) multiplies their quadratic by (2x+1), with at least 4 of the 6 terms correct
A1 correct unsimplified expansion, e.g. 2x³+14x²+24x+x²+7x+12 oe
A1 2x³ + 15x² + 31x + 12 cao
Answer: 2x³ + 15x² + 31x + 12

Question 6

M1 expands two of the three brackets correctly, e.g. (x-3)(x+4) = x²+x-12 oe
M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
A1 cso - fully correct working shown leading to x³ + 2x² - 11x - 12, with no errors
Answer: x³ + 2x² - 11x - 12 (shown)

Question 7

(a) M1 expands two of the three brackets correctly, e.g. (x+2)(x+3) = x²+5x+6 oe
(a) M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
(a) A1 cso - fully correct working shown leading to x³ + 10x² + 31x + 30, with no errors
(a) Answer: x³ + 10x² + 31x + 30 (shown)
(b) M1 substitutes x = 1 into their expression from part (a) or into (x+2)(x+3)(x+5) directly
(b) A1 72 cao (cm³)
(b) Answer: 72 cm³

Question 8

M1 expands (x+4)(x-1)(x+2) fully, or compares coefficients of x, to form an equation for k
A1 k = 2 cao
Answer: k = 2

Question 9

M1 writes (x+2)³ as (x+2)(x+2)(x+2) and expands (x+2)(x+2) = x²+4x+4 oe
M1 (dep) multiplies their quadratic by (x+2), with at least 4 of the 6 terms correct
A1 correct unsimplified expansion, e.g. x³+2x²+4x²+8x+4x+8 oe
A1 x³ + 6x² + 12x + 8 cao
Answer: x³ + 6x² + 12x + 8

Question 10

M1 recognises (x+5)(x-5) as the difference of two squares
A1 x² - 25 oe correct
M1 (dep) multiplies (x²-25) by (x+6), with at least 3 of the 4 terms correct
A1 x³ + 6x² - 25x - 150 cao
Answer: x³ + 6x² - 25x - 150

Question 11

M1 writes (x+4)³ as (x+4)(x+4)(x+4) and expands (x+4)(x+4) = x²+8x+16 oe
M1 (dep) multiplies their quadratic by (x+4), with at least 4 of the 6 terms correct
A1 correct unsimplified expansion, e.g. x³+4x²+8x²+32x+16x+64 oe
A1 x³ + 12x² + 48x + 64 cao
Answer: x³ + 12x² + 48x + 64

Question 12

(a) M1 expands two of the three brackets correctly, e.g. (x-2)(x+5) = x²+3x-10 oe
(a) M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
(a) A1 cso - fully correct working shown leading to x³ - 19x + 30, including the x² term correctly cancelling to 0, with no errors
(a) Answer: x³ - 19x + 30 (shown)
(b) B1 0 (ft from their part (a) cubic, must be consistent with an x³ + 0x² + ... form)
(b) Answer: 0

Question 13

M1 expands (x+4)(x-3) correctly to get x²+x-12 oe
M1 (dep) multiplies by (x+a) and compares coefficients to form an equation in a
M1 correct equation, e.g. a + 1 = 3, or -12a = -24
A1 a = 2 cao
Answer: a = 2

Question 14

M1 expands (x+1)(x+4) correctly to get x²+5x+4 oe
M1 (dep) multiplies their quadratic by (3-x), taking care with signs, with at least 4 of the 6 terms correct
A1 correct unsimplified expansion, e.g. 3x²+15x+12-x³-5x²-4x oe
A1 -x³ - 2x² + 11x + 12 oe cao
Answer: -x³ - 2x² + 11x + 12

Question 15

M1 expands (3x-1)(x-4) correctly to get 3x²-13x+4 oe
A1 3x² - 13x + 4 correct
M1 (dep) multiplies their quadratic by (x+2), with at least 4 of the 6 terms correct
A1 correct unsimplified expansion, e.g. 3x³-13x²+4x+6x²-26x+8 oe
A1 3x³ - 7x² - 22x + 8 cao
Answer: 3x³ - 7x² - 22x + 8

Question 16

(a) M1 expands two of the three brackets correctly, e.g. (x-5)(x+2) = x²-3x-10 oe
(a) M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
(a) A1 x³ - 2x² - 13x - 10 cao
(a) Answer: x³ - 2x² - 13x - 10
(b) M1 substitutes x = 0 into f(x) = (x-5)(x+2)(x+1), or into their expanded expression from part (a)
(b) A1 f(0) = -10 cao
(b) Answer: f(0) = -10
(c) B1 (0, -10) stated (ft from part (b))
(c) Answer: (0, -10)

Question 17

M1 expands (x-2)(2x+3) correctly to get 2x²-x-6 oe
A1 2x² - x - 6 correct
M1 (dep) multiplies their quadratic by (3x+1), with at least 4 of the 6 terms correct
A1 correct unsimplified expansion, e.g. 6x³-3x²-18x+2x²-x-6 oe
A1 6x³ - x² - 19x - 6 cao
Answer: 6x³ - x² - 19x - 6

Question 18

M1 expands (x+2)(x+3)(x+4) correctly to x³+9x²+26x+24 oe
M1 expands (x-2)(x-3)(x-4) correctly to x³-9x²+26x-24 oe
M1 (dep on both) subtracts the two cubics correctly, with the x³ and x terms cancelling
A1 cso - fully correct working shown leading to 18x² + 48, with no errors
Answer: 18x² + 48 (shown)

Question 19

(a) M1 expands two of the three brackets correctly, e.g. (x+4)(x-1) = x²+3x-4 oe
(a) M1 (dep) multiplies their quadratic by the remaining bracket, with at least 4 of the 6 terms correct
(a) A1 cso - fully correct working shown leading to x³ + 6x² + 5x - 12, with no errors
(a) Answer: x³ + 6x² + 5x - 12 (shown)
(b) B1 sets V = 30 and rearranges x³+6x²+5x-12=30 to x³+6x²+5x-42=0 with no errors
(b) Answer: x³ + 6x² + 5x - 42 = 0 (shown)
(c) M1 substitutes x = 2 into x³+6x²+5x-42 and evaluates to 0, e.g. 8+24+10-42=0
(c) A1 dimensions 6 cm, 1 cm and 5 cm (ft from x = 2), all three stated
(c) Answer: x = 2 satisfies the equation (8 + 24 + 10 - 42 = 0); dimensions are 6 cm, 1 cm and 5 cm

Question 20

B1 correctly identifies that n, n+1 and n+2 are three consecutive integers, so at least one is a multiple of 2 and exactly one is a multiple of 3
M1 expands n(n+1)(n+2) = n³ + 3n² + 2n oe, or otherwise sets up the product to support a valid argument
A1 cso - complete, correctly reasoned argument that the product must be divisible by both 2 and 3, and hence by 6, with no gaps
Answer: n(n+1)(n+2) = n³ + 3n² + 2n, which is always a multiple of 6 (proof required)

Expanding Triple Brackets: Fluency and Exam Drill - Worksheets, Questions and Revision