Created by T. Madas
ALGEBRAIC
FRACTIONS
Exam Questions
Created by T. Madas
Created by T. Madas
Question 1
(**)
Express
3x − 4
2
x − 5x − 6
−
2
,
x−6
as a single fraction in its simplest form.
C3C ,
Question 2
1
x +1
(**)
Show that
x ( x − 6 ) − ( x − 1)( x + 5 )
=k,
1 − 2x
where k is an integer to be found.
k =5
Created by T. Madas
Created by T. Madas
Question 3
(**)
Solve the equation
x+
9 15
= , x ≠ 0.
x 2
x = 3 ,6
2
Question 4
(**+)
Solve the equation
x
3
+4= , x ≠ 0.
x−2
x
x = 1, 6
5
Created by T. Madas
Created by T. Madas
Question 5
(**+)
Show clearly that
1+
x −8
2
−
x + 2x − 8
2
x− p
≡
,
x+4 x−q
stating the value of each of the integer constants, p and q .
C3G , p = 3 , q = 2
Question 6
(**+)
Given that
2 x3 + x − 2
2
x +1
≡ Ax + B +
Cx + D
x2 + 1
,
use polynomial division, or another appropriate method, to find the value of each of the
constants A , B , C and D .
C3B ,
Created by T. Madas
( A, B, C , D ) = ( 2, 0, −1, −2 )
Created by T. Madas
Question 7
(**+)
1−
1
3
+ 2
, x ≠ 2 , x ≠ 1.
x−2 x −x−2
Write the above algebraic expression as a single simplified fraction
C3M ,
Question 8
x
x +1
(**+)
x4 + 1
2
x +1
≡ Ax 2 + B +
C
2
x +1
.
Find the value of each of the constants A , B and C .
A = 1 , B = −1 , C = 2
Created by T. Madas
Created by T. Madas
Question 9
(***)
Show that
( x2 − 5)( x2 + 3) − 15 ( x2 − 2) + 1 ≡ x + A ,
( x2 + 5x + 4)( x2 − 3x + 2) x + B
stating the value of each of the constants A and B .
A = −4 , B = −2
Question 10
(***)
Solve the equation
2
13
+ 2
= 1, x ≠ 3 , x ≠ 7 .
x − 3 x + 4 x − 21
x = −8,6
Created by T. Madas
Created by T. Madas
Question 11
(***)
Solve the equation
9
−
2
x + 15 x + 54
2
1
=
, x ≠ −6 , x ≠ −9 .
x+9 x+6
x = −4
Question 12
(***)
Given that
2 x3 + x 2 − 4 x + 1
2
x + x−2
≡ Ax + B +
C
,
x+D
use polynomial division, or another appropriate method, to find the value of each of the
constants A , B , C and D .
C3N ,
Created by T. Madas
( A, B, C , D ) = ( 2, −1,1, 2 )
Created by T. Madas
Question 13
(***)
Solve the equation
x + 11
2
−
2x − 5x − 3
x −1
+ 2 = 0 , x ≠ − 1 , x ≠ 3.
2
x−3
x =1
Question 14
(***)
Find, in exact surd form, the roots of the equation
x 2 + 3x
x2 + 5x + 6
=
2 x2 − x − 1
x2 + 8x − 9
, x ≠ −3 , x ≠ 1 .
x = 2± 2
Created by T. Madas
Created by T. Madas
Question 15
(***)
5x
(
)(
x2 + 2 4 x2 + 3
≡
Ax + B
) (
x2 + 2
Cx + D
+
) (
4 x2 + 3
)
,
where A , B , C and D are constants.
Determine the value of each of the constants A , B , C and D .
A = −1 , B = 0 , C = 4 , D = 0
Question 16
(***)
Show clearly that the expression
1−
1
3
+ 2
,
x−2 x −x−2
can be written in the form
x+a
,
x+b
where a and b are integer constants to be found.
C3O , a = 0 , b = 1
Created by T. Madas
Created by T. Madas
Question 17
(***)
f ( x) ≡
4x −1
3
−
− 2 , x ∈ » , x ≠ 1, x ≠ 1 .
2
2 ( x − 1) 2 ( x − 1)( 2 x − 1)
Show clearly that
f ( x) ≡
k
, x∈», x ≠ 1 ,
2
2x −1
where k is an integer to be found.
C3I , k = 3
Question 18
(***)
Given that
4 x 4 + 4 x3 − 23x 2 − 4
2
x + x−6
≡ Ax 2 + Bx + C −
C
,
x+E
find the value of each of the constants A , B , C , D and E .
C3P ,
Created by T. Madas
( A, B, C , D, E ) = ( 4,0,1,1,3)
Created by T. Madas
Question 19
(***)
Show clearly that
( 3x − 1)( 2 x + 3) − 2 ( 4 x − 1)  ( 3x − 1)
≡ ax 2 + bx + c ,
3x + 1
where a , b and c are integers to be found.
C3Q , a = 6 , b = −5 , c = 1
Question 20
(***+)
Show clearly that
x3 − 3 x + 2
( x − 1) ( x + 2)
2
≡
x+ A
,
x+B
stating the value of each of the constants A and B .
A = −1 , B = 1
Created by T. Madas
Created by T. Madas
Question 21
(***+)
Find the solution of the equation
2 x 2 + x − 1 2 3x − 1
+ =
.
x x −1
x2 − x
x = 3, x ≠ 1
Created by T. Madas
Created by T. Madas
Question 22
(***+)
a) Find the solutions of the equation
2
11
+
= 1 , y ≠ −8 , y ≠ 0 .
y y ( y + 8)
b) Hence, or otherwise, solve the equation
2
11
+
=1, x ≠ − 3 , x ≠ 5 .
16
16
16 x − 5 (16 x − 5)(16 x + 3)
x = −9,3 , x = − 1 , 1
4 2
Question 23
(***+)
Show clearly that
x
x+4
−
x −3 x+3 ≡ 2 .
x+6
x2 − 9
proof
Created by T. Madas
Created by T. Madas
Question 24
(***+)
f ( x) ≡ x −
12
2
x + 2x − 3
+
3
, x ∈ » , x ≠ −3 , x ≠ 1 .
x −1
Show clearly that
f ( x) ≡
x 2 + 3x + 3
, x ∈ » , x ≠ −3 , x ≠ 1 .
x+3
C3H , proof
Question 25
(***+)
Show clearly that
1
1
− 2
x+a
12 3x
≡
.
1 1
1
x
+
b
+
+
12 4 x 6 x 2
where a and b are integers to be found
C3R ,
Created by T. Madas
x−2
x +1
Created by T. Madas
Question 26
(****)
3x + 1 3x
2
−
+
= 0 , x ≠ −1 , y ≠ 2 .
x +1 y − 2 x +1
Show that the above expression represents a straight line and give its equation.
y = x+2
Question 27
(****)
Show clearly that
9 x3 − 9 x 2 − x + 1
3x − 1
≡ ax + b ,
x −1
where a and b are integers to be found.
a =1 , b =1
Created by T. Madas
Created by T. Madas
Question 28
(****)
f ( x) ≡
6 x 2 − 21x + 17
( x − 3)( x − 1)2
, x ∈ » , x ≠ 3, x ≠ 1.
a) Express f ( x ) into partial fractions.
g ( x) ≡
( x − 12 )( x + 1) ,
( x − 3)( x − 1)2
x ∈ » , x ≠ 3, x ≠ 1.
b) Use the result of part (a) to express g ( x ) into partial fractions.
No credit will be given in this part by repeating the method used in part (a)
SYN-R , f ( x ) ≡
4
2
1
+
−
x − 1 x − 3 ( x − 1)2
Created by T. Madas
f ( x) ≡
4
3
1
−
−
x − 1 x − 3 ( x − 1) 2
Created by T. Madas
Question 29
(****+)
Simplify the following expression
1
1
2x
4 x3
−
−
−
,
1 − x 1 + x 1 + x2 1 + x4
giving the final answer as a single algebraic fraction in its simplest form.
8 x7
1 − x8
Question 30
(****+)
Solve the following equation.
2+ 2x
2
x + 2 x +1
+
2− 2x
2
x − 2 x +1
= 2, x∈».
SYN-T , x = ±1
Created by T. Madas
Created by T. Madas
Question 31
(*****)
f ( x) ≡
4
4
x +1
, x∈».
Express f ( x ) into partial fractions over the set of real numbers.
SPX-T , f ( x ) ≡
2+ 2x
2
x + 2 x +1
Created by T. Madas
+
2− 2x
2
x − 2 x +1
=2
Created by T. Madas
Question 32
(*****)
Solve the following rational equation, over the set of real numbers.
5x
2 x2 − 7 x + 3
+
4 x 2 − 37 x + 13 6 x 2 − 22 x − 21
1
+
=
.
2
2
3x + 2
2 x − 11x + 5
3x − 7 x − 6
You may ignore non finite solutions.
SPX-Q , x = 7
Created by T. Madas
Created by T. Madas
Question 33
(*****)
f ( x) ≡
3x 4 + x3 + 2 x 2 + x − 2
( x + 1) x5
, x ∈ » , x ≠ 0 , x ≠ −1
Express f ( x ) into partial fractions in their simplest form.
C4T , f ( x ) ≡ −
Created by T. Madas
2
x
5
+
3
x
4
−
1
x
3
+
2
x
2
+
1
1
−
x x +1
Created by T. Madas
Question 34
(*****)
f ( x) ≡
( 2 x + 1)2 ,
4
x ( x + 1)
x ∈ » , x ≠ 0 , x ≠ −1
Express f ( x ) into partial fractions in their simplest form.
SP-X , f ( x ) ≡
1
1
1
3
1
−
−
+
−
2
3
x x + 1 ( x + 1)
( x + 1) ( x + 1)4
Created by T. Madas