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
q(x) h(x) q(x) * h(x)p(x) - g(x) = p(x) * h(x) - g(x) * q(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
THANKS FOR READING THIS AT Math Capsule.
