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Factor Theorem
Question Paper 1
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 1
Algebra
Factor Theorem
Question Paper 1
Time Allowed:
83 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
(a)
The polynomial f(x) is given by f(x)= x3ͻ͜ͻ᷊x + 15.
(i)
Use the Factor Theorem to show that x + 3 is a factor of f(x).
(2)
(ii)
Express f(x) in the form (x + 3)(x2 + px + q), where p and q are integers.
(2)
(b)
A curve has equation y = x4ͻ͜ͻⅎx2 + 60x + 7.
(i)
Find
.
(3)
(ii)
Show that the x-coordinates of any stationary points of the curve satisfy the equation
x3ͻ͜ͻ᷊x + 15 = 0
(1)
(iii)
Use the results above to show that the only stationary point of the curve occurs when
xͻⱣͻ᷉᷄͜
(2)
(iv)
Find the value of
when xͻⱣͻ᷉᷄͜
(3)
(v)
Hence determine, with a reason, whether the curve has a maximum point or a
minimum point when xͻⱣͻ᷉᷄͜
(1)
(Total 14 marks)
Page 1 of 3
3
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)
4
The polynomial p(x) is given by
p(x) = x3 + 2x2 – 5x – 6
(a)
(i)
Use the Factor Theorem to show that x + 1 is a factor of p(x).
(2)
(ii)
Express p(x) as the product of three linear factors.
(3)
(b)
Verify that p(0) > p(1).
(2)
(c)
Sketch the curve with equation y = x3 + 2x2 – 5x – 6, indicating the values where the curve
crosses the x-axis.
(3)
(Total 10 marks)
5
(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 2 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)
6
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)
7
The polynomial p(x) is given by p(x) = x3 – 13x – 12.
(a)
Use the Factor Theorem to show that x + 3 is a factor of p(x).
(2)
(b)
Express p(x) as the product of three linear factors.
(3)
(Total 5 marks)
Page 3 of 3