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Algebraic FractionsAlgebraic Fractions
Addition
bd
bcad
d
c
b
a
xx 2
31Simplify
)2(
)(3)2(1
xx
xx
)2(
32
xx
xx
)2(
22
xx
x
Addition
bd
bcad
d
c
b
a
y
x
x 2
3Simplify
)2(
))(()2(3
yx
xxy
xy
xy
2
6 2
Subtraction
bd
bcad
d
c
b
a
p
t
p 2
2Simplify
)2)((
))(()2)(2(
pp
tpp
22
4
p
ptp
22
)4(
p
tp
p
t
2
)4(
Can you see an alternative, quicker method here?
Subtraction
bd
bcad
d
c
b
a
4
33
2
12
xxSimplify
)4)(2(
)33(2)12(4
xx
8
6648
xx
8
22 x
8
)1(2
x4
4
1x
Multiplication
bd
ac
d
c
b
a
5
5
3
x
xxSimplify
)5(3
)5)((
x
xx
)5(3
52
x
xx
Multiplication
bd
ac
d
c
b
a
62
5
15
3
x
xSimplify
)62)(15(
)5)(3(
x
x
)3)(2)(15(
)5)(3(
x
xFactorising (2x-6)Factorising (2x-6)
3
)2)(3(
1
6
1
Division
bc
ad
d
c
b
a
Simplify
14
4
7
2 yx
y
x
4
14
7
2
)4)(7(
)14)(2(
y
x
2
2
y
x
Example: 2
4
1
2
xx
)2)(1(
xx
The denominators don’t share any factors
)2(2 x )1(4 x
)2)(1(
6
xx
x
Example :2)1(
4
1
2
x
x
x
The 1st denominator is a factor of the 2nd
)2)(1(
26
xx
x
)2)(1(
)13(2
xx
x
2)1(
x
)1(2 x x4
The denominators share a factor
Example :
23
3
34
222
xxxx
We have to find the factors
)2(
3
)3(
2
xx)1( x )1( x
)2)(3)(1(
xxx42 x 93 x
)2)(3)(1(
5
xxx
x
)2)(3(
xx)1( x)2(2 x )3(3 x
)2)(1(
12
xx
x
Example :6
3
2
1222
xx
x
xx
x
)3)(2)(1(
xxx352 2 xx )34( 2 xx
)3)(2)(1(
62
xxx
xx
)3)(2)(1(
xxx
)3)(2(
3
xx
x
)3)(2)(1(
xxx352 2 xx 342 xx
)3)(2)(1(
)2)(3(
xxx
xx1
1
1
x
)3)(12( xx )1)(3( xx
)3)(2)(1(
xxx
EquationsOnce you know how to simplify algebraic fractions, youCan solve equations containing them
For example
743
2xx
35
2
2
1
xx
Click an equation to see it solved
743
2xx
Solve
7)4)(3(
))(3()2)(4(
xx
712
38
xx
712
5x
845 x5
84 x
Solve 35
2
2
1
xx
3)5)(2(
)2)(2()1)(5(
xx
310
4255
xx 310
97
x
3097 x
217 x
3x
Short Quiz :Express each of the following as a single fraction in its simplest form.
4
5
3
2
xx1.
)4)(3(
)3(5)4(2
xx
xx
)4)(3(
15582
xx
xx
)4)(3(
77
xx
x
)4)(3(
)1(7
xx
x
2. )3)(2(
32
3
3
xx
x
x )2)(3(
)32()2(3
xx
xx
)2)(3(
3263
xx
xx
)2)(3(
3
xx
x
2
1
x
6
23
65
1222
xx
x
xx
x3.
)3)(2(
23
)3)(2(
12
xx
x
xx
x
)2)(3)(2(
)2)(23()2)(12(
xxx
xxxx
)2)(3)(2(
443232 22
xxx
xxxx
)2)(3)(2(
65 2
xxx
xx
)2)(3)(2(
)1)(65(
xxx
xx
4. xxx
x
22
4
1
3
)1(
4
)1)(1(
3
xxxx
x
)1)(1(
)1(4)3(
xxx
xxx
)1)(1(
4432
xxx
xxx
)1)(1(
42
xxx
xx
