RATIONAL EXPRESSION
Rational Expression : If m and n are two integers, n ¹ 0 then the quotient m/n is not necessarily an integer. A fractional number is defined as the quotient m/n of two integers m and n where n ¹ 0. Similarly if p(x) and q(x) are two polynomials such that q(x) ¹ 0 then p(x)/q(x) need not be a polynomial. It is called a rational expression. Every polynomial is a rational expression, need not to be a polynomial.
For example
x2 – 3x + 2 is a rational x2 – 1
expression, because both the numerator and denominator are polynomials.
Rational expressions in lowest terms : Let p(x) and q(x) be two polynomials having integral coefficients,
then the rational expression
p(x)
q(x)
said to be in its lowest terms if the g.c.d. of p(x) and q(x) is one.
Example : The rational expression
(x – 3)(x – 5) is not in its lowest terms, (x – 3)(x + 2)
because g.c.d. of its numerator and denominator in not one. The g.c.d. of its numerator and denominator is (x – 3).
Algorithm to reduce a rational expression in its lowest terms : The following algorithm can help to explain.
a given rational expression p(x) in its lowest terms.
q(x)
Step 1. First find out the factors of the two polynomials p(x) and q(x).
Step 2. Calculate g.c.d. of p(x) and q(x).
Step 3. If g.c.d. of p(x) is one, then
the given rational expression p(x) is in
q(x)
its lowest terms. Otherwise, divide the numerator p(x) and denominator q(x) by the g.c.d. of p(x) and q(x).
Step 4. The rational factor obtained in step 3 is in its lowest terms.
Example : Reduce each of the following rational expression to its lowest
terms :
2×2 – 5x + 3 . x2 – 3x + 2
Solution : Let p(x) = 2×2 – 5x + 3 = (x – 1) (2x – 3) And, q(x) = x2 – 3x + 2 = (x – 2) (x – 1)
Clearly, g.c.d. [p(x),q(x)] = x – 1.
\ 2×2 – 5x + 3 = (x – 1)(2x – 3) = 2x – 3
x2 – 3x + 2 (x – 1)(x – 2) x – 2
The given rational expression in its
lowest term is =
2x – 3
x – 2
Addition of rational expressions :
If p(x) and r(x) are two rational
q(x) s(x)
expressions having the same denominator, then their sum is
p(x) + r(x) = p(x) + r(x)
q(x) q(x) q(x)
Example : The sum of the two rational expressions x + 5 and x + 2
x + 3 x – 2
Solution : x + 5 and
x + 2
x – 3 x – 2
= (x + 5) (x – 2) + (x + 2) (x – 3) (x – 3)(x – 2)
= (x2 – 3x – 10) + (x2 – x – 6)
(x2 – 5x + 6)
= p(x)s(x)+q(x)r(x)
q(x)s(x)
PROPERTIES OF SUM OF RATIONAL EXPRESSIONS
Closure property : The sum of two rational expressions is always a rational expression and possesses closure property.
Commutative law : Sum of rational expressions is commutative. For any two
rational expressions p(x) and r(x) ,
q(x) s(x)
we have
p(x) + r(x) = r(x) + p(x) .
q(x) s(x) s(x) q(x)
Existence of additive identity : The rational expression
0 = 0 is the additive identity i.e., 1
p(x) 0 p(x) 0 p(x)+ = = + .
q(x) 1 q(x) 1 q(x)
for all rational expressions p(x) .
q(x)
Existance of additive inverse : For every rational expression p(x), then exists
q(x)
a rational expression (additive inverse) -p(x) such that
q(x)
p(x) + -p(x) = 0 = -p(x) + p(x)
q(x) q(x) q(x) q(x)
Example : The additive inverse of a rational expression
x2 – 5x is -(x2 – 5x) = 5x – x2
2x – 3 2x – 3 2x – 3
Subtraction of rational expressions :
If p(x) and r(x)are two rational expressions, then the subtraction of
q(x) s(x)
r(x)
from
p(x) is denoted by p(x)
– r(x)
and is defined as the sum of
p(x)
s(x) q(x) q(x) s(x) q(x)
and the additive inverse of
r(x) , i.e., s(x)
p(x) – r(x) = p(x) + [-r(x)]
q(x) s(x) q(x) s(x)
= p(x).s(x) – q(x).r(x) q(x).s(x)
The difference of
x – 3
- x + 2 is
x + 4 x + 3
= (x – 3)(x + 3) – (x + 2)(x + 4) (x + 4)(x + 3)
= (x2 – 9) – (x2 + 6x + 8) x2 + 7x + 12
= – 6x – 17 x2 – 7x + 12
= -(6x + 17) x2 + 7x + 12
Multiplication of rational expressions :
If p(x) and r(x) are two rational expressions, then we define
q(x) s(x)
p(x) r(x) p(x).r(x)
q(x)
C =
s(x) q(x).s(x)
Example : If
p(x) = x + 3 and r(x)
= x – 4 , then
q(x) x + 1 s(x) x – 1
p(x) r(x) (x + 3)(2 – 4) x2 – x – 12
q(x)
C = = . s(x) (x + 1)(x – 1) x2 – 1
PROPERTIES OF MULTIPLICATION OF RATIONAL EXPRESSIONS
Closure property : The multiplication of two rational expressions is a rational expression and shows closure property.
Commutative property : The multiplication of rational expressions shows com- mutative property i.e.
p(x) r(x) r(x) p(x) p(x) r(x)
q(x) C
s(x) =
s(x) C
q(x) for any two rational expressions
q(x)
and s(x) .
Associative property : The multiplication of rational expression is
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associative i.e., (p(x)
r(x) u(x)
|
C C
= p(x) C (r(x)
u(x)
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C fro any three
rational expressions p(x) , u(x)
and r(x) .
q(x) v(x) s(x)
Existence of multiplicative identity : The rational expression
1 is the multiplicative identity i.e., p(x)
C 1 = p(x) = 1 C
p(x) for any rational
1 q(x) 1 q(x) 1 q(x)
expression p(x) .
q(x)
Existence of multiplicative inverse : For every non-zero rational expression
p(x) there exist a rational expression
q(x)
such that p(x)
q(x)
C
q(x) p(x) q(x) p(x)
= 1 =
q(x)
C
p(x) .
p(x) q(x)
Here q(x) is called the multiplicative inverse or reciprocat of
p(x) .
p(x) q(x)
Distributive property : The multiplication of rational expressions is distributive over their addition i.e., for any three rational expressions
p(x) , r(x) and u(x), we have
p(x) r(x)
C
+ u(x) =
p(x) r(x)
C +
p(x)
u(x)
C
q(x) s(x) v(x) q(x) s(x) v(x) q(x) s(x) q(x) v(x)
where p(x) ¹ 0, s(x) ¹ 0 and v(x) ¹ 0.
The following are some results of the properties of addition and multiplication of rational expressions.
(1) p(x) C q(x) = 0 Þ at least one of p(x) and q(x) is zero (2) p(x) + q(x) = 0 Þ q(x) = – p(x)
(3) p(x) C q(x) = p(x) C r(x) and p(x) ¹ 0, then q(x) = r(x)
(4) p(x) C q(x) = r(x) and p(x) ¹ 0
Þ q(x) =
r(x)
p(x)
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(5) – (-p(x)) = p(x)
(6) 1 -1 = p(x) and [p(x)]-1 = 1 p(x) p(x)
(7) [-p(x)] C q(x) = – [p(x) C q(x)] = p(x) C [-q(x)] (8) [- p(x)] C [-q(x)] = p(x) q(x)
Division of rational expressions :
Let p(x)
be a rational expression and let
r(x) be a non-zero rational expression
q(x) s(x)
such that r(x) ¹ 0 and s(x) ¹ 0, then we define
p(x)
÷ s(x) =
p(x)
C
r(x) .
q(x) r(x) q(x) s(x)
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p(x) r(x) r(x)
be a rational expression and let be a non-zero rational s(x) s(x)
expression so that r(x) ¹ 0 and s(x) ¹ 0,
then then we define p(x) ÷
r(x)
= p(x) C
r(x) .
q(x) s(x) q(x) s(x) In other words, dividing a polynomial
p(x)
by the reciprocal of
r(x) .
q(x) s(x)
Example :
x2 + 1 ÷ x2 – 1 = x2 + 1 2x – 1 =
2x3 – x2 + 2x – 1
x – 2 2x – 1 x – 2 C
x2 – 3 x3 – 2x2 – 3x + 6
Properties of Rational Numbers and Rational Expressions RATIONAL NUMBERS
- The result of addition of two rational numbers is a rational
- Addition has the commutative and associative
- All rational numbers has additive
- All integers are considered as rational
- Multiplication of all rational numbers result in rational
- Product has the commutative as well as associative
- Product has distributive
- Every rationall number which is non-zero has a
RATIONAL EXPRESSIONS
- The result of addition of two rational expressions is a rational
- Addition is commutative as well as
- They have additive
- All polynomials are treated as rational
- Multiplication of two rational expressions is a rational
- Product has the commutative and associative
- Multiplication has distributive
- Every rational expression which is non-zero has a reciprocal
