Complex Fractions
Frank Ma © 2011
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction.
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction.
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication,
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
AB C
D*
flip
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
AB C
D*
flip
Example A. Simplify
4x2y9xy2
6
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
AB C
D*
flip
Example A. Simplify
4x2y9xy2
6
=
flip
4x2y9 xy2
6
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
AB C
D*
flip
Example A. Simplify
4x2y9xy2
6
=
flip
4x2y9 xy2
63
2
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
AB C
D*
flip
Example A. Simplify
4x2y9xy2
6
=
flip
4x2y9 xy2
63
2x
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
AB C
D*
flip
Example A. Simplify
4x2y9xy2
6
=
flip
4x2y9 xy2
63
2x
y
Complex FractionsA fraction with its numerator or the denominator containing fraction(s) is called a complex fraction. In other words, a complex fraction is a fractional expression divided by another fractional expression.We want to simplify complex fractions to regular fractions.The “easy” complex fractions are just regular divisions of of a fraction by another fraction. In this case just flip the denominator then simplify the multiplication, i.e.
ABCD
AB C
D*
flip
Example A. Simplify
4x2y9xy2
6
=
flip
4x2y9 xy2
6 =3
2x
y
8x3y
Complex FractionsWe give two methods for simplifying general complex fractions.
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem.
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction.
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions.
Example B. Simplify
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions. x
x + 1 + 1
1 – xx – 1
Example B. Simplify
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions. x
x + 1 + 1
1 – xx – 1 x
x + 1 + 1
1 – xx – 1
Example B. Simplify
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions. x
x + 1 + 1
1 – xx – 1 x
x + 1 + 1
1 – xx – 1
=
xx + 1 +
– xx – 1
11
11
Example B. Simplify
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions. x
x + 1 + 1
1 – xx – 1 x
x + 1 + 1
1 – xx – 1
=
xx + 1 +
– xx – 1
11
11
=
x + (x + 1)x + 1
Example B. Simplify
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions. x
x + 1 + 1
1 – xx – 1 x
x + 1 + 1
1 – xx – 1
=
xx + 1 +
– xx – 1
11
11
=
x + (x + 1)x + 1(x – 1) – x
x – 1
Example B. Simplify
Complex FractionsWe give two methods for simplifying general complex fractions. The first method is to reduce the problem to an “easy” problem. We do this by combining the numerator into one fraction and the denominator into one fraction. We can use cross multiplication for combining two fractions. x
x + 1 + 1
1 – xx – 1 x
x + 1 + 1
1 – xx – 1
=
xx + 1 +
– xx – 1
11
11
=
x + (x + 1)x + 1(x – 1) – x
x – 1
=
2x + 1x + 1
–1x – 1
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
2x + 1x + 1
–1x – 1
Therefore,
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
2x + 1x + 1
–1x – 1
Therefore,
= (2x + 1)(x + 1)
(x – 1) (–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
2x + 1x + 1
–1x – 1
Therefore,
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1) (–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1) (–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1) (–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1) (–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1) (–1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1) (–1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1)
This is OK since is 1.12121
(–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12
6
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1)
Distribute the multiplication.
(–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12
6 12
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1)
Distribute the multiplication.
(–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12
6 12 2
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1)
Distribute the multiplication.
(–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12
6 12 2
12 4 3
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1)
Distribute the multiplication.
(–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12=
6 12 2
12 4 318 + 12 – 1012 – 8 + 9
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1) (–1)
Complex Fractions
xx + 1 + 1
1 – xx – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to the numerator and the denominator of the complex fraction.
Example C. Simplify
32 + 1
1 –
– 56
23
34+
The LCD of 23
32
56
34, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.32 + 1
1 –
– 56
23
34+ (
( ))
12
12=
6 12 2
12 4 318 + 12 – 1012 – 8 + 9 = 20
13
2x + 1x + 1
–1x – 1
= (2x + 1)(x + 1)
(x – 1) = – (2x + 1)(x + 1)
(x – 1) (–1)
Complex Fractionsx2
+ 1
– 1 Example D. Simplify x
y
y2
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
y2
x2
xy y2 ,
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
1
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
y21
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
y2
y2y
1
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2
= (x – y)(x + y)y(x + y)
To simplify this, put it in the factored form .
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2 To simplify this, put it in the factored form .
= (x – y)(x + y)y(x + y)
Complex Fractionsx2
+ 1
– 1 Example D. Simplify
The LCD of
xy
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2
xy y2 ,
x2
+ 1
– 1
xy
y2
((
)) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2 To simplify this, put it in the factored form .
= (x – y)(x + y)y(x + y)
= x – y y
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
3x + 2
xx – 2
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2 (
( ))
(x – 2)(x + 2)
(x – 2)(x + 2)
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2 (
( ))
(x – 2)(x + 2)
(x – 2)(x + 2)
(x + 2)
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2 (
( ))
(x – 2)(x + 2)
(x – 2)(x + 2)
(x + 2) (x – 2)
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2 (
( ))
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)x(x + 2) – 3(x – 2) Expand and put the fraction
in the factored form.
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2 (
( ))
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)x(x + 2) – 3(x – 2) Expand and put the fraction
in the factored form. = x2 + 2x + 3x – 6
x2 + 2x – 3x + 6
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2 (
( ))
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)x(x + 2) – 3(x – 2) Expand and put the fraction
in the factored form. = x2 + 2x + 3x – 6
x2 + 2x – 3x + 6
= x2 + 5x – 6x2 – x + 6
Complex Fractionsx
x – 2
–
+
3x + 2
Example E. Simplify
The LCD of
3x + 2
xx – 2
xx – 2
xx – 2
3x + 2
3x + 2 , , , is (x – 2)(x + 2).
Multiply the LCD to top and bottom of the complex fraction.
xx – 2
–
+
3x + 2
3x + 2
xx – 2 (
( ))
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)x(x + 2) – 3(x – 2) Expand and put the fraction
in the factored form. = x2 + 2x + 3x – 6
x2 + 2x – 3x + 6
= x2 + 5x – 6x2 – x + 6
This is the simplified.= (x + 6)(x – 1)x2 – x + 6
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Use the crossing method.
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Use the crossing method.2
x – 1
–
+
3x + 2
3x + 2
1x – 2
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Use the crossing method.2
x – 1
–
+
3x + 2
3x + 2
1x – 2
=
2(x + 2) + 3(x – 1)(x – 1)(x + 2)
2(x + 3) – 3(x – 2)(x – 2)(x + 2)
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Use the crossing method.2
x – 1
–
+
3x + 2
3x + 2
1x – 2
=
2(x + 2) + 3(x – 1)(x – 1)(x + 2)
2(x + 3) – 3(x – 2)(x – 2)(x + 2)
=
5x +1(x – 1)(x + 2) – x + 12
(x – 2)(x + 2)
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Use the crossing method.2
x – 1
–
+
3x + 2
3x + 2
1x – 2
=
2(x + 2) + 3(x – 1)(x – 1)(x + 2)
2(x + 3) – 3(x – 2)(x – 2)(x + 2)
flip the bottom fraction
=
5x +1(x – 1)(x + 2) – x + 12
(x – 2)(x + 2)
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Use the crossing method.2
x – 1
–
+
3x + 2
3x + 2
1x – 2
=
2(x + 2) + 3(x – 1)(x – 1)(x + 2)
2(x + 3) – 3(x – 2)(x – 2)(x + 2)
flip the bottom fraction
=
5x +1(x – 1)(x + 2) – x + 12
(x – 2)(x + 2)
=(5x +1)
(x – 1)(x + 2) (– x + 12)(x – 2)(x + 2)
Complex Fractions2
x – 1
–
+
3x + 2
Example F. Simplify
3x + 2
1x – 2
Use the crossing method.2
x – 1
–
+
3x + 2
3x + 2
1x – 2
=
2(x + 2) + 3(x – 1)(x – 1)(x + 2)
2(x + 3) – 3(x – 2)(x – 2)(x + 2)
flip the bottom fraction
=
5x +1(x – 1)(x + 2) – x + 12
(x – 2)(x + 2)
=(5x +1)
(x – 1)(x + 2) (– x + 12)(x – 2)(x + 2)
=(5x + 1)(x – 2)(x – 1)(–x + 12)
Complex FractionsEx. A. Simplify.
1.
2x2y54y2 2.
34x2
3x2 3. 4x2
2x3 4.
12x2
5y4xy15
3
+ 1
–
13
45.
2 3
5
–
–
23
66.
4 314
4
+
–
16
57.
11 15712
1
+ 1
– 2
2y
x8.1
+ 2
1 –
1x2
xy9.1
+
–
1x
y10.1x
1y
2
–
–
4x2
y11.2x
4y2
2
–
–
4x2
y12.2x
4y2
2x + 1 + 1
1 – 1x – 1
13.
Complex Fractions1
2x + 1 – 2
3 – 1x + 2
14.
2x – 3 – 1
2 – 12x + 1
15.
2x + 3 –
+ 1x + 3
16.
3x + 2
3x + 2
–2 2x + 1 –
+ 3x + 4
17.
1x + 4
22x + 1
2 x + 2 –
+ 2x + 5
18.
13x – 1
3x + 2
4 2x + 3 –
+ 3x + 4
19.
33x – 2
53x – 2
–5 2x + 5 –
+ 3–x + 4
20.
22x – 3
62x – 3
23 + 2
2 –
– 16
23
12+
21.
12 – + 56
23
14–
22.34
32 +
Complex Fractions
23.
2x – 1
–
+
3x + 3
xx + 3
xx – 1
24.
3x + 2
–
+
3x + 2
xx – 2
xx – 2
25.2
x + h – 2x
h26.
3x – h– 3
xh
27.2
x + h – 2x – h
h28.
3x + h–
h
3x – h