REDUCING EACH RATIONAL EXPRESSIONS TO LOWEST TERMS

Definition of rational expression :

An expression is called a rational expression if it can be written in the form of 

p(x)/q(x)

(where p(x) and q(x) are polynomials and q(x) ≠ 0)

Reduction of rational expressions :

A rational expression p(x)/q(x) is said to be in its lowest form if 

GCD [p(x), q(x)]  =  1

To reduce a rational expression to its lowest form, follow the steps given below. 

Step 1 : 

Factorize the numerator and the denominator

Step 2 :

If there are common factors in the numerator and denominator, cancel them.

Step 3 :

The resulting expression will be a rational expression in its lowest form.

Note :

Sometimes it may be necessary to use algebraic identities.

Example 1 :

Express in simplest form.

x/(x2)

Solution :

=  x/(x2)

=  1/x

So, the answer is 1/x.

Example 2 :

Express in simplest form.

Solution :

By using cross multiplication, we get

So, the answer is (x - 3)/2.

Example 3 :

Express in simplest form.

Solution :

So, the answer is x - 2.

Example 4 :

Express in simplest form.

Solution :

So, the answer is (x - 3)/(x + 2).

Example 5 :

Express in simplest form.

The expression (x2y2 - 9)/(3 - xy) is equivalent to 

(A)  -1     (B)  1/(3 + xy)     (C)  -(3 + xy)     (D)  3 + xy

Solution :

So, the answer is -(xy + 3)

Example 6 :

Express in simplest form.

Solution :

So, the answer is (x + 3)/2.

Example 7 :

Expressed as a fraction in lowest terms, 

(x2 - x - 2)/(x2 - 4), x ≠ ±2 is equivalent to

(A)  x/(x+2)    (B)  (x-1)/(x-2)   (C)  (x+1)/(x+2)   

(D) (-x-2)/(-4)

Solution :

So, the answer is (x + 1)/(x + 2).

Example 8 :

What is (6x2 - 4x)/(2x2 - 4x) reduced to simplest term ?

(A)  (3x-2)/(x+2)    (B)  (3x+2)/(x-2)   (C)  (3x+2)/(x+2)   

(D) (3x-2)/(x-2)

Solution :

By taking common factor, we get

So, the answer is (3x - 2)/(x - 2).

Example 9 :

What is (x2 + x - 20)/(4 - x) expressed in simplest form ?

(A)  -x - 5    (B)  x + 5  (C)  5 - x   (D)  x - 5

Solution :

So, the answer is -x - 5.

Example 10 :

Expressed in simplest form, 

is equivalent to

(A)  n + 1    (B)  n  (C)  n - 1   (D)  (n - 1)/(n + 1)

Solution :

By cross multiplication, we get

So, the answer is n - 1.

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