Handout - Fractions and Rational Expressions - Northeastern University

Handout - Fractions and Rational Expressions

Algebraic Rules for Fractions and Rational Expressions

a, b, c, d may be numbers or variable expressions.

a c a+c

+ =

b b

b

Adding and Subtracting

requires

has Common Denominator

a d c b a¡¤d+c¡¤d

a c

+ = ¡¤ + ¡¤ =

b d

b d d b

b¡¤d

a Common Denominator

a c

a¡¤c

¡¤ =

b d b¡¤d

Multiplying

just multiply tops/bottoms

a/b a d a ¡¤ d

= ¡¤ =

c/d

b c

b¡¤c

Dividing - ¡±flip¡± bottom

a¡¤c a

=

b¡¤c

b

Reducing Fractions

cancel common terms

a¡¤c+b¡¤c a+b

=

d¡¤c

d

a¡¤c

a

=

b¡¤c+d¡¤c b+d

1

Reducing Fractions and Rational Expressions

Technique: Factor and Cancel

(Cancel Multiplicative Factors Only)

Example 1a: Simplify and Reduce:

6

8

6

2¡¤3

=

8

2¡¤4

Factor

6

2¡¤3



=

8

2¡¤4



Cancel common factor

=

3

4

Example 1b: Simplify and Reduce:

Answer

7+8

7 + 28

7+8

7 + 28

cannot cancel 7¡¯s!

7+8

15

=

7 + 28

35

Add

=

3¡¤5

5¡¤7

Factor

=

3 ¡¤ 5

5¡¤7



Cancel

=

3

7

Answer

2

Example 1c: Simplify and Reduce:

6x ? 9

6x + 12

6x ? 9

6x + 12

cannot cancel 6x!

3 ¡¤ (2x ? 3)

6x ? 9

=

6x + 12

3 ¡¤ (2x + 4)

=

=

Factor

3¡¤



(2x ? 3)

3 ¡¤ (2x + 4)



Cancel

2x ? 3

2x + 4

Answer

Example 1d: Simplify and Reduce:

9x + 6

15x + 10

9x + 6

3 ¡¤ (3x + 2)

=

15x + 10

5 ¡¤ (3x + 2)

Factor



3 ¡¤

(3x

+2)



=





5 ¡¤

(3x

+2)





=

Cancel Common Factor

3

5

Answer

3

8x5 + 6x4

Example 1e: (x terms included) Simplify and Reduce:

10x5 + 20x3

Observe 1: 2 divides all the integers (8, 6, 10, 20)

Observe 2: x3 divides all the xn terms (x5 , x4 , x5 , x3 )

2 ¡¤ (4x5 + 3x4 )

8x5 + 6x4

=

10x5 + 20x3

2 ¡¤ (5x5 + 10x3 )

Factor Common Integers

(4x5 + 3x4 )

=

2 ¡¤ (5x5 + 10x3 )

Cancel Common Factor

2 ¡¤

=

4x5 + 3x4

5x5 + 10x3

=

x3 ¡¤ (4x2 + 3x)

x3 ¡¤ (5x2 + 10)

=

=

x3 ¡¤ (4x2 + 3x)

x3 ¡¤ (5x2 + 10)

4x2 + 3x

5x2 + 10

Factor Common xn

Cancel Common Factor

Answer

4

Simplify and Reduce the following Fractions/Rational Expressions

a)

48

18

b)

48

30

c)

156

247

d)

114

102

e)

30 + 6

27 + 1

f)

160 + 5

173 + 7

g)

12x + 15

27x ? 9

h)

24x ? 36

4x ? 16

32x4 ? 24x3

i)

20x4 ? 44x3

k)

84x3 ? 126x2

j)

77x3 + 63x2

45x ? 40

9x ? 8

Answers a) 38 ; b) 85 ; c)

l)

12

19 ;

d)

19

17 ;

e) 79 ; f)

11

12 ;

g)

4x+5

9x?3 ;

5

8x ? 3

8x ? 3

h)

6x?9

x?4 ;

i)

6?8x

11?5x ;

j)

6(2x?3)

11x+9 ;

k) 5; l) 1;

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