Rational Expressions – Core 4 Revision
1.
It is given that
f(x) = 2x3 – x2 – 8x + 4.
(a)
Use the factor theorem to show that (x – 2) and (2x – 1) are factors of f(x).
(3)
(b)
Write f(x) as a product of three linear factors.
(2)
(c)
Hence, by writing x = 2t, solve the equation
23t+1 – 22t –2t+3 + 4 = 0.
(3)
(Total 8 marks)
2.
It is given that
f(x) = 4x3 – 17x2 + 16x – 3.
(a)
Use the factor theorem to show that (x – 3) and (4x – 1) are factors of f(x).
(3)
(b)
Express f(x) as a product of three linear factors.
(2)
(c)
(i)
Show that f(2t) = 23t+2 – 17(22t) + 2t+4 – 3.
(2)
(ii)
Hence find the non-integer root of the equation
23t+2 –17(22t) + 2 t+4 – 3 = 0,
giving your answer to three significant figures.
(4)
(Total 11 marks)
3.
The polynomial f(x) is defined by f(x) = 2x3  7x2 + 13.
(a)
Use the Remainder Theorem to find the remainder when f (x) is divided by (2x  3).
(2)
(b)
The polynomial g(x) is defined by g(x) = 2x3  7x2 + 13 + d, where d is a constant.
Given that (2x  3) is a factor of g(x), show that d = 4.
(2)
(c)
Express g(x) in the form (2x  3)(x 2 + ax + b).
(2)
(Total 6 marks)
South Wolds Comprehensive School
1
4.
(a)
(i)
Show that ( x + 2) is a factor of p(x) = 2x3 – x2 – 8x + 4.
(1)
(ii)
Hence factorise p(x) completely into linear factors.
(3)
(b)
Sketch the graph of y = 2x3 – x2 – 8x + 4, and hence solve the inequality
2x3 – x2 – 8x + 4 > 0.
(3)
(Total 7 marks)
5.
(a)
Given that (2x – 1) is a factor of p(x) = 6x3 – kx2 – 6x + 8, use the factor theorem to show
that k = 23.
(2)
(b)
Express p(x) as the product of three linear factors.
(3)
(Total 5 marks)
6.
(a)
(i)
Express
3x  5
B
in the form A 
A, where A and B are integers.
x 3
x 3
(2)
(ii)
Hence find

3x  5
dx.
x 3
(2)
(b)
(i)
Express
6x  5
4 x  25
2
in the form
Q
P

, where P and Q are integers.
2x  5 2x  5
(3)
(ii)
Hence find
6x  5
 4x
2
 25
dx.
(3)
(Total 10 marks)
7.
(a)
Find the remainder when 2x2 + x  3 is divided by 2x + 1.
(2)
(b)
Simplify the algebraic fraction
2x 2  x  3
x2  1
(3)
(Total 5 marks)
8.
The polynomial p(x) is given by
p(x) = (x + 3)(x – 2)(x – 4)
(a)
Find the remainder when p(x) is divided by (x + 1).
(b)
(i)
Express
(2)
70
A
B
C


in the form
.
( x  3)( x  2)( x  4)
x3 x2 x4
(3)
(Total 5 marks)
South Wolds Comprehensive School
2
9.
(a)
The polynomial p(x) is defined by p(x) = 6x3 – 19x2 + 9x + 10.
(i)
Find p(2).
(1)
(ii)
Use the Factor Theorem to show that (2x + 1) is a factor of p(x).
(3)
(iii)
Write p(x) as the product of three linear factors.
(2)
(b)
Hence simplify
3x 2  6 x
.
6 x  19 x 2  9 x  10
3
(2)
(Total 8 marks)
10.
(a)
Given that (x + 2) is a factor of
p(x) = 6x3 + kx2 – 9x + 2
show that k = 7.
(2)
(b)
Find the value of p  1  and hence show that (2x – 1) is a factor of p(x).
2
(2)
(c)
Express p(x) as a product of three linear factors.
(2)
(d)
Hence find the values of , in radians, in the interval 0 <  < 2 for which
6 sin3  + 7 sin2  – 9 sin  + 2 = 0
(6)
(Total 12 marks)
11.
(a)
Find the remainder when 2x3 – x2 + 2x – 2 is divided by 2x – 1.
(2)
(b)
3
2
Given that 2 x  x  2 x  2 = x2 + a + b , find the values of a and b.
2x  1
2x  1
(4)
(Total 6 marks)
12.
A polynomial is given by p(x) = 6x3  7x2  x + 2:
(a)
Find the value of p 
(b)
Use the factor theorem to show that (x  1) is a factor of p(x).
(2)
(c)
Write p(x) as a product of three linear factors.
(3)
(d)
Hence find the values of , in radians, in the interval  <  < , for which
1
2
.
6 cos3   7 cos2   cos  + 2 = 0
(1)
(6)
(Total 12 marks)
South Wolds Comprehensive School
3