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6.5 Expanding Triple Brackets

EDEXCEL 1MA1 · Calculator allowed · about 85 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

When you fully expand three linear brackets, each containing a term in x, such as (x + a)(x + b)(x + c), what is the highest power of x in the fully simplified answer?

A) x
B) x²
C) x³
D) x⁴

(Total for Question 1 is 1 mark)

Expand and simplify (x + 2)(x + 3).
(This is a warm-up for expanding three brackets later in this worksheet.)

(Total for Question 2 is 2 marks)

Which of the following is the correct expansion of (x + 1)(x + 1)(x + 1)?

A) x³ + 1
B) x³ + 3x + 1
C) x³ + 3x² + 3x + 1
D) x³ + 3x² + 1

(Total for Question 3 is 1 mark)

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

(Total for Question 4 is 3 marks)

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

(Total for Question 5 is 3 marks)

Show that (x + 4)(x - 4)(x + 2) = x³ + 2x² - 16x - 32

(Total for Question 6 is 3 marks)

Expand and simplify (x + 7)(x - 7)(x + 3).

(Total for Question 7 is 3 marks)

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

(Total for Question 8 is 4 marks)

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

(Total for Question 9 is 4 marks)

A student expands (x - 3)(x + 2)(x - 1) and writes the final answer as x³ - 2x² - 5x - 6.
The student has made one error.
Work out the correct expansion, and state which term in the student's answer is wrong.

(Total for Question 10 is 3 marks)

A cuboid has dimensions (x + 1) cm, (x + 2) cm and (x + 5) cm, where x > 0.

Diagram NOT accurately drawn

(a)Show that the volume, V cm³, of the cuboid is given by V = x³ + 8x² + 17x + 10(3)

(b)Use your calculator to work out the volume of the cuboid when x = 3.5 cm. Give your answer to 3 significant figures.(2)

(Total for Question 11 is 5 marks)

The volume of a fish tank in the shape of a cuboid is given by V = (x + 3)(x + 1)(x - 2) cm³, where x > 2.

Diagram NOT accurately drawn

(a)Show that V = x³ + 2x² - 5x - 6(3)

(b)The volume of the tank is 144 cm^3. Show that x = 5 fits this volume, and use it to find the three dimensions of the tank.(3)

(Total for Question 12 is 6 marks)

Expand and simplify (x + 5)³
Give your answer in the form x³ + ax² + bx + c

(Total for Question 13 is 4 marks)

Expand and simplify (x - 3)³
Give your answer in the form x³ + ax² + bx + c

(Total for Question 14 is 4 marks)

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

(Total for Question 15 is 4 marks)

When (x + 3)(x - 2)(2x + k) is expanded and simplified, the coefficient of x² is 5.
Find the value of k.

(Total for Question 16 is 3 marks)

(x + 1)(x - 2)(x + 4) = x³ + ax² + bx + c for all values of x.

(a)Find the values of a, b and c.(3)

(b)Hence write down the three roots of the equation x³ + ax² + bx + c = 0.(2)

(Total for Question 17 is 5 marks)

Given that (x + 2)(x + a)(x - 1) = x³ + 4x² + x - 6 for all values of x, find the value of a.

(Total for Question 18 is 3 marks)

Prove algebraically that, for any positive integer n,
(n + 1)(n + 2)(n + 3) - n(n + 1)(n + 2) is always a multiple of 6.

(Total for Question 19 is 4 marks)

Mark scheme · 6.5 Expanding Triple Brackets

Question 1

B1 C selected (cao)
Answer: C (x³)

Question 2

M1 at least 3 of the 4 terms correct from the grid/FOIL expansion, e.g. x² + 3x + 2x + 6 (oe)
A1 x² + 5x + 6 (cao)
Answer: x² + 5x + 6

Question 3

B1 C selected (cao)
Answer: C (x³ + 3x² + 3x + 1)

Question 4

M1 expands any two of the three brackets correctly, e.g. x² + 3x + 2 (oe)
M1 multiplies their quadratic by the remaining linear 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 any two of the three brackets correctly, e.g. x² + x - 6 (oe)
M1 multiplies their quadratic by the remaining linear bracket, with at least 4 of the 6 terms correct
A1 x³ - 7x + 6 (cao)
Answer: x³ - 7x + 6

Question 6

M1 uses the difference of two squares on the first pair of brackets, x² - 16 (oe)
M1 multiplies (x² - 16) by (x + 2), with at least 3 of the 4 terms correct
A1 x³ + 2x² - 16x - 32 shown correctly with no errors (cso)
Answer: x³ + 2x² - 16x - 32 (shown)

Question 7

M1 uses the difference of two squares on the first pair of brackets, x² - 49 (oe)
M1 multiplies (x² - 49) by (x + 3), with at least 3 of the 4 terms correct
A1 x³ + 3x² - 49x - 147 (cao)
Answer: x³ + 3x² - 49x - 147

Question 8

M1 expands (2x + 1)(x + 2) correctly, 2x² + 5x + 2 (oe)
M1 correct method to multiply their quadratic by (x - 3)
A1 at least 4 of the 6 resulting terms correct before final simplifying, ft their quadratic
A1 2x³ - x² - 13x - 6 (cao)
Answer: 2x³ - x² - 13x - 6

Question 9

M1 expands (x + 1)(x - 2) correctly, x² - x - 2 (oe)
M1 multiplies their quadratic by (x + 3), with at least 4 of the 6 terms correct
A1 x³ + 2x² - 5x - 6 (oe, ft) before doubling
A1 2x³ + 4x² - 10x - 12 (cao)
Answer: 2x³ + 4x² - 10x - 12

Question 10

M1 complete correct method to expand all three brackets, e.g. via (x-3)(x+2) = x²-x-6 then x(x²-x-6) - 1... or equivalent grid method
A1 correct expansion x³ - 2x² - 5x + 6 (cao)
B1 identifies the constant term as the error, stating it should be +6, not -6 (since (-3) x (2) x (-1) = +6)
Answer: Correct expansion: x³ - 2x² - 5x + 6; the student's constant term is wrong (should be +6, not -6)

Question 11

(a) M1 expands (x + 1)(x + 2) correctly, x² + 3x + 2 (oe)
(a) M1 multiplies their quadratic by (x + 5), with at least 4 of the 6 terms correct
(a) A1 x³ + 8x² + 17x + 10 shown correctly with no errors (cso)
(a) Answer: x³ + 8x² + 17x + 10 (shown)
(b) M1 substitutes x = 3.5 into V = x³ + 8x² + 17x + 10 (or into (x+1)(x+2)(x+5))
(b) A1 awrt 210 cm³ (units not required for the mark but expected in the answer space)
(b) Answer: 210 cm³ (3 s.f.)

Question 12

(a) M1 expands (x + 3)(x + 1) correctly, x² + 4x + 3 (oe)
(a) M1 multiplies their quadratic by (x - 2), with at least 4 of the 6 terms correct
(a) A1 x³ + 2x² - 5x - 6 shown correctly with no errors (cso)
(a) Answer: x³ + 2x² - 5x - 6 (shown)
(b) M1 substitutes x = 5 into V = x³ + 2x² - 5x - 6 (or into (x+3)(x+1)(x-2)) and evaluates
(b) A1 correctly shows V = 144 when x = 5 (cso)
(b) A1 states all three dimensions: 8 cm, 6 cm and 3 cm
(b) Answer: V(5) = 144, confirmed; dimensions are 8 cm, 6 cm and 3 cm

Question 13

M1 writes (x + 5)³ as (x + 5)(x + 5)(x + 5), or expands (x+5)(x+5) correctly, x² + 10x + 25 (oe)
M1 multiplies their quadratic by the third (x + 5), with at least 4 of the 6 terms correct
A1 x³ + 15x² + 75x + 125 (oe unsimplified form)
A1 fully simplified x³ + 15x² + 75x + 125 (cao)
Answer: x³ + 15x² + 75x + 125

Question 14

M1 writes (x - 3)³ as (x - 3)(x - 3)(x - 3), or expands (x-3)(x-3) correctly, x² - 6x + 9 (oe)
M1 multiplies their quadratic by the third (x - 3), with at least 4 of the 6 terms correct
A1 x³ - 9x² + 27x - 27 (oe unsimplified form)
A1 fully simplified x³ - 9x² + 27x - 27 (cao)
Answer: x³ - 9x² + 27x - 27

Question 15

M1 expands (2x - 1)(x + 3) correctly, 2x² + 5x - 3 (oe)
M1 correct method to multiply their quadratic by (3x - 2)
A1 at least 4 of the 6 resulting terms correct, ft their quadratic
A1 6x³ + 11x² - 19x + 6 (cao)
Answer: 6x³ + 11x² - 19x + 6

Question 16

M1 expands (x + 3)(x - 2) correctly, x² + x - 6 (oe)
M1 expands (x² + x - 6)(2x + k) in terms of k and identifies the x² coefficient as (k + 2), oe
A1 k = 3 (cao)
Answer: k = 3

Question 17

(a) M1 expands (x + 1)(x - 2) correctly, x² - x - 2 (oe)
(a) M1 multiplies their quadratic by (x + 4), with at least 4 of the 6 terms correct
(a) A1 a = 3, b = -6, c = -8, all correct (cao)
(a) Answer: a = 3, b = -6, c = -8
(b) B1 two of x = -1, x = 2, x = -4 correct (ft from the original factorised form)
(b) B1 all three of x = -1, x = 2, x = -4 correct
(b) Answer: x = -1, x = 2, x = -4

Question 18

M1 expands (x + 2)(x + a)(x - 1) in terms of a, e.g. via (x+2)(x+a) = x² + (a+2)x + 2a
M1 forms and uses a correct equation by comparing coefficients (e.g. constant term -2a = -6, or coefficient of x²: a + 1 = 4)
A1 a = 3 (cao)
Answer: a = 3

Question 19

M1 identifies (n+1)(n+2) as a common factor of both terms
A1 factorises correctly to (n+1)(n+2)[(n+3) - n], oe
A1 simplifies to 3(n+1)(n+2), oe
B1 complete reasoning: (n+1) and (n+2) are consecutive integers so one must be even, making (n+1)(n+2) even, so 3(n+1)(n+2) is a multiple of 3 x 2 = 6
Answer: Proven: (n+1)(n+2)(n+3) - n(n+1)(n+2) = 3(n+1)(n+2), which is always a multiple of 6

Expanding Triple Brackets - Worksheets, Questions and Revision