Simplifying Basic

Rational Expressions

Prepared by: Sandra

SanvictoresLeslie Limjuco

Duane Rodriguez

The objective is to be able to simplify a rational expression

Another objective is to know how to factor the numerator and denominator and to,

Divide out the common factors

Polynomial – an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).

Rational expression - a fraction in which the numerator and/or the denominator are polynomials.

Domain of a rational expression –  the set of all possible values of the variables.

Reduced form – a rational expression in which the numerator and denominator have no factors in common.

Divide out the common factors

Factor the numerator and denominator and then divide the common factors

Step 1 – Identify any factors which are common to both the numerator and the denominator.

55 7

xx( )

The numerator and denominator have a common factor.The common factor is the five.

Step 2 – Divide out the common factors.The fives can be divided since 5/5 = 1The x remains in the numerator.The (x-7) remains in the denominator

55 7

xx( )

xx 7

Factor the numerator.Factor the denominator.Divide out the common factors.Write in simplified form.

3 91 2

2

3x x

x

Step 1: Look for common factors to both terms in the numerator.

3 912

2

3x x

x 3 is a factor of both 3 and

9.X is a factor of both x2 and x.Step 2: Factor the numerator.

3 91 2

2

3x x

x 3 3

12 3x xx

( )

Step 3: Look for common factors to the terms in the denominator and factor.

3 91 2

2

3x x

x

The denominator only has one term. The 12 and x3 can be factored. The 12 can be factored into 3 x 4.The x3 can be written as x • x2.3 9

1 2

2

3x x

x 3 3

3 4 2x xx x

( )

Step 4: Divide out the common factors. In this case, the common factors divide to become 1.

3 33 4 2x x

x x( )

Step 5: Write in simplified form.

xx

34 2

Simplify the following rational expressions.

1 92 4

2

2. x yzxyz

2 34 32. a

a a

3 3 157 1 02. x

x x

4 2 1 51 2

2

2. x xx x

5 14 35 2 11 2 30 18

2

2. x xx x

92 4

2

2x yzxyz

3 33 8

x x y zx y z z

33 x y zx y z

38xz

1 38xz

38xz

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

aa a

34 32 a

a a

3

3 1( )( )Factor the numerator and denominator

aa

33

11a

1 11a 1

1a

Divide out the common factors.

Write in simplified form.

3 157 102x

x x

3 55 2

( )( )( )

xx x

xx

55

32x

1 3

2x 3

2x

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

x xx x

2

22 15

12 ( )( )

( )( )x xx x

5 34 3

xx

33

xx

54

1 xx

54 x

x

54

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

1 4 35 211 2 30 1 8

2

2x xx x

7 2 5 36 2 6 3

2

2( )( )x xx x

( )( )

2 5 32 6 3

2

2x xx x

76

1 76

76

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.