Save My Exams! – The Home of Revision
For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/
Remainder theorem
Question Paper 1
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 1
Algebra
Remainder theorem
Question Paper 1
Time Allowed:
82 minutes
Score:
/68
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
1
The polynomial p(x) is given by
p(x) = x3ͻ͜ͻ᷊x2ͻ͜ͻ᷉x + 18
(a)
Use the Remainder Theorem to find the remainder when p(x) is divided by x + 1.
(2)
(b)
(i)
Use the Factor Theorem to show that xͻ͜ͻ᷉ͻⱳᴠͻԛͻᴓԛԝ₸ꜜꜟͻꜜᴓͻꜝӾx).
(2)
(ii)
Express p(x) as a product of linear factors.
(3)
(c)
Sketch the curve with equation y = x3ͻ͜ͻ᷊x2ͻ͜ͻ᷉x + 18, stating the values of x where the
curve meets the x-axis.
(3)
(Total 10 marks)
2
The polynomial p(x) is given by p(x) = x 3 + cx 2 + dxͻ͜ͻ᷇᷈ԒͻᴤⱲᴒꜟᴒͻc and d are constants.
(a)
When p(x) is divided by xͻԑͻ᷈Ԓͻ₸Ⱳᴒͻꜟᴒꞌԛⱳꜛᴑᴒꜟͻⱳᴠͻ᷇᷾᷆᷄͜
Show that 2cͻ͜ͻd + 65 = 0.
(3)
(b)
Given that xͻ͜ͻ᷉ͻⱳᴠͻԛͻᴓԛԝ₸ꜜꜟͻꜜᴓͻp(x), find another equation involving c and d.
(2)
(c)
By solving these two equations, find the value of c and the value of d.
(3)
(Total 8 marks)
3
(a)
(i)
Sketch the curve with equation y = x(x – 2)2.
(3)
(ii)
Show that the equation x(x – 2)2 = 3 can be expressed as
x3 – 4x2 + 4x – 3 = 0
(1)
(b)
The polynomial p(x) is given by p(x) = x3 – 4x2 + 4x – 3.
(i)
Find the remainder when p(x) is divided by x + 1.
(2)
(ii)
Use the Factor Theorem to show that x – 3 is a factor of p(x).
(2)
(iii)
Express p(x) in the form (x – 3)(x2 + bx + c), where b and c are integers.
(2)
Page 1 of 3
(c)
Hence show that the equation x(x – 2)2 = 3 has only one real root and state the value of
this root.
(3)
(Total 13 marks)
4
The polynomial p(x) is given by p(x) = x3 – 2x2 + 3.
(a)
Use the Remainder Theorem to find the remainder when p(x) is divided by x – 3.
(2)
(b)
Use the Factor Theorem to show that x + 1 is a factor of p(x).
(2)
(c)
(i)
Express p(x) = x3 – 2x2 + 3 in the form (x + 1)(x2 + bx + c), where b and c are
integers.
(2)
(ii)
Hence show that the equation p(x) = 0 has exactly one real root.
(2)
(Total 8 marks)
5
The polynomial p(x) is given by
p(x) = x3 + 7x2 + 7x – 15
(a)
(i)
Use the Factor Theorem to show that x + 3 is a factor of p(x).
(2)
(ii)
Express p(x) as the product of three linear factors.
(3)
(b)
Use the Remainder Theorem to find the remainder when p(x) is divided by x – 2.
(2)
(c)
(i)
Verify that p(–1) < p(0).
(1)
(ii)
Sketch the curve with equation y = x3 + 7x2 + 7x – 15, indicating the values where the
curve crosses the coordinate axes.
(4)
(Total 12 marks)
6
(a)
The polynomial p(x) is given by p(x) = x3 – x + 6.
(i)
Find the remainder when p(x) is divided by x – 3.
(2)
Page 2 of 3
(ii)
Use the Factor Theorem to show that x + 2 is a factor of p(x).
(2)
(iii)
Express p(x) = x3 – x + 6 in the form (x + 2)(x2 + bx + c), where b and c are integers.
(2)
(iv)
The equation p(x) = 0 has one root equal to –2. Show that the equation has no other
real roots.
(2)
(b)
The curve with equation y = x3 – x + 6 is sketched below.
The curve cuts the x-axis at the point A(–2, 0) and the y-axis at the point B.
(i)
State the y-coordinate of the point B.
(1)
(ii)
Find
.
(5)
(iii)
Hence find the area of the shaded region bounded by the curve y = x3 – x + 6 and
the line AB.
(3)
(Total 17 marks)
Page 3 of 3