Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
Factor Theorem
Question Paper 1
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 4
Algebra
Factor Theorem
Question Paper 1
Time Allowed:
85 minutes
Score:
/71
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
Q1.The polynomial f(x) is defined by f(x)= 4x3 − 11x − 3.
(a)
Use the Factor Theorem to show that (2x + 3) is a factor of f(x).
(2)
(b)
Write f(x) in the form (2x + 3)(ax2 + bx + c), where a, b and c are integers.
(2)
(c)
(i)
Show that the equation 2 cos 2θ sin θ + 9 sin θ + 3 = 0 can be written as
4x3 − 11x − 3 = 0, where x = sin θ.
(3)
(ii)
Hence find all solutions of the equation 2 cos 2θ sin θ + 9 sin θ + 3 = 0 in
the interval 0° < θ < 360°, giving your solutions to the nearest degree.
(4)
(Total 11 marks)
Q2.(a)
Use the Factor Theorem to show that 4x − 3 is a factor of
16x3 + 11x − 15
(2)
(b)
Given that x = cos θ, show that the equation
27 cos θ cos 2θ + 19 sin θ sin 2θ − 15 = 0
can be written in the form
16x3 + 11x − 15 = 0
(4)
Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
(c)
Hence show that the only solutions of the equation
27 cos θ cos 2θ + 19 sin θ sin 2θ − 15 = 0
are given by cos θ =
.
(4)
(Total 10 marks)
Q3.
(a)
(i)
The polynomial f (x) is defined by f (x) = 9x3 + 18x2 – x – 2.
Use the Factor Theorem to show that 3x + 1 is a factor of f (x).
(2)
(ii)
Express f (x) as a product of three linear factors.
(3)
(iii)
Simplify
.
(3)
(b)
When the polynomial 9x3 + px2 – x – 2 is divided by 3x – 2, the remainder is –
4.
Find the value of the constant p.
(2)
(Total 10 marks)
The polynomial f(x) is defined by f(x) = 4x3 – 13x + 6.
Q4.
(a)
Find f(–2).
Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
(1)
(b)
Use the Factor Theorem to show that 2x – 3 is a factor of f(x).
(2)
(c)
Simplify
(4)
(Total 7 marks)
The polynomial f(x) is defined by f(x) = 15x3 + 19x2 – 4.
Q5.
(a)
(i)
Find f(–1).
(1)
(ii)
Show that (5x – 2) is a factor of f(x).
(2)
(b)
Simplify
giving your answer in a fully factorised form.
(5)
(Total 8 marks)
Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
Q6.
A botanist is investigating the rate of growth of a certain species of toadstool.
She observes that a particular toadstool of this type has a height of 57 millimetres at
a time 12 hours after it begins to grow.
She proposes the model
, where A is a constant, for the height
h millimetres of
the toadstool, t hours after it begins to grow.
(a)
Use this model to:
(i)
find the height of the toadstool when t = 0;
(1)
(ii)
show that A = 60, correct to two significant figures.
(2)
(b)
Use the model
(i)
to:
show that the time T hours for the toadstool to grow to a height of 48
millimetres is given by
T = a ln b
where a and b are integers;
(3)
(ii)
show that
;
(3)
(iii)
find the height of the toadstool when it is growing at a rate of 13
millimetres per hour.
(1)
(Total 10 marks)
Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
Q7.
(a)
The polynomial f(x) is defined by f(x) = 8x3 + 6x2 – 14x – 1.
Find the remainder when f(x) is divided by (4x – 1).
(2)
(b)
The polynomial g(x) is defined by g(x) = 8x3 + 6x2 – 14x + d.
(i)
Given that (4x – 1) is a factor of g(x), find the value of the constant d.
(2)
(ii)
Given that g(x) = (4x – 1)(ax2 + bx + c), find the values of the integers a,
b and c.
(3)
(Total 7 marks)
Q8.
(a)
(i)
The polynomial f(x) is defined by f(x) = 4x3 – 7x – 3.
Find f(–1).
(1)
(ii)
Use the Factor Theorem to show that 2x + 1 is a factor of f(x).
(2)
Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
(iii)
Simplify the algebraic fraction
.
(3)
(b)
The polynomial g(x) is defined by g(x) = 4x3 – 7x + d. When g(x) is divided by
2x + 1, the remainder is 2. Find the value of d.
(2)
(Total 8 marks)