RATIONAL EXPRESSIONS math capsule

Today we are going to study about  rational expressions,

RATIONAL EXPRESSIONS

An expression of the form p(x) / q (x) , where p(x) and q(x) are polynomials and q(x) ≠ 0 is called a rational expression.

* In the rational expression p(x) / q(x) , p(x) is called the numerator and q(x) is called the denominator of the rational expression.

* Every polynomial can be said to be rational expression. Since p(x) can always be written as p(x) / 1 .

* Every rational expression need not be a polynomial .

WORKING RULE TO REDUCE THE GIVEN RATIONAL EXPRESSION IN ITS LOWEST TERM

STEP 1
Firstly, factorise both the polynomials  p(x) and q(x).
STEP 2
Find the HCF of p(x) and q(x) . If HCF of p(x) and q(x) is one , the rational expression p(x) / q(x) is in its lowest terms.
STEP 3
If HCF is not equal to 1. Then divide both p(x) and q(x) by their and the rational expression obtained in the lowest term.

SHORTCUT--------

Place p(x) and q(x) as p(x) / q(x) and factorise , cancel out common factor the resulting polynomial is in the lowest term.

Q. The lowest term of an expression

    a³ - b³                             is

 a² + ab + b²

Rational expression =     a³ - b³                             

                                     a² + ab + b²

= (a - b)( a² + ab + b²) /  (a² + ab + b²)

= (a - b)

OPERATIONS ON RATIONAL EXPRESSIONS

(1) ADDITION OF RATIONAL EXPRESSIONS

If p(x) / q(x) and g(x) / h(x) are two rational expressions , then their sum is 

{p(x)/q(x)}  +   {g(x)/h(x)}  =  p(x) * h(x) + q(x) * g(x)   

                                                                     q(x) * h(x)                                                                        
           

Additive inverse of p(x) / q(x) is   - p(x)/q(x) .

'0' is the additive identity.

(2) SUBTRACTION OF RATIONAL EXPRESSIONS 

If p(x)/q(x) and g(x)/h(x) are two rational expressions then their subtraction is 

p(x)  -  g(x)   =  p(x) * h(x)  - g(x) * q(x)

                    q(x)      h(x)              q(x) * h(x)

(3) MULTIPLICATION OF RATIONAL EXPRESSIONS

If p(x) / q(x) and g(x) / h(x) are two rational expression , then their product is given by 

p(x) * g(x)   =   p(x) * g(x)

           q(x)    h(x)        q(x) * h(x)

Multiplicative Inverse of p(x)/q(x) is q(x)/p(x) .

'1' is multiplicative identity.

(4) DIVISION OF RATIONAL EXPRESSIONS

If p(x)/q(x) and g(x)/h(x) are two rational expressions, then their division is given by 

p(x)  ÷ g(x)  = p(x) * h(x)

          q(x)     h(x)      q(x) * g(x)

It is the product of p(x) / q(x) and the reciprocal of g(x) / h(x) .i.e. h(x) / g(x) .

EXERCISES

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