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# Rational Expressions

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In this page we are going to discuss about Rational Expressions concept . Expression is a finite combination of symbols that are well formed and rational expressions is that where the numerator and the denominator or both of them are polynomials. A rational number is a few number that can be printed in the form p/q, there are p is specified the integer and q is also specified the integers and q ? 0. Rational expression: If p(x) and q(x) are two polynomials, q(x) [!=] 0, then the quotient [(p(x))/(q(x))] is called a rational expression. * p(x) is known as numerator and q(x) is known as the denominator of the Rational Expression * [(p(x))/(q(x)) ] need not be a polynomial Since a rational number is of the form p/q, we can say that a rational expression is also of the form p/q, but since it is an expression, it contains variables along with numbers . ["Rational expression" = ("algebraic expressions(s)")/("algebraic expressions(s)")] Steps to reduce a given rational expression to its lowest terms: * Factorize each of the two polynomials p(x) and q(x) * Find highest common divisor of p(x) and q(x) * If h.c.f. = 1 , then the given rational expression [(p(x))/(q(x))] is in its lowest terms * If h.c.f [!=] 1, then divide the numerator p(x) and denominator q(x) by the h.c.f of p(x) and q(x) * The rational expression obtained in Step III or Step IV is in its lowest terms Example: Reduce the rational expression [(x^2-5x-6)/(x^2+3x+2)] to its lowest terms Solution: Let p(x) = x2 -5x-6 = (x-6)(x+1) Let q(x) = x2 + 3x +2 = (x+2)(x+1) Clearly h.c.f of p(x), q(x) = (x+1) [(x^2-5x-6)/(x^2+3x+2) = [(x-6)(x+1)]/[(x+2)(x+1)]] [= (x-6)/(x+2)] (cancel out h.c.f = (x+1))
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Rational Expressions
In this page we are going to discuss about Rational Expressions concept . Expression is a finite combination
of symbols that are well formed and rational expressions is that where the numerator and the denominator or
both of them are polynomials. A rational number is a few number that can be printed in the form p/q, there are
p is specified the integer and q is also specified the integers and q ? 0.
Rational expression:
If p(x) and q(x) are two polynomials, q(x) [!=] 0, then the quotient [(p(x))/(q(x))] is called a rational expression.
* p(x) is known as numerator and q(x) is known as the denominator of the Rational Expression
* [(p(x))/(q(x)) ] need not be a polynomial
Since a rational number is of the form p/q, we can say that a rational expression is also of the form p/q, but
since it is an expression, it contains variables along with numbers .
["Rational expression" = ("algebraic expressions(s)")/("algebraic expressions(s)")]
Steps to reduce a given rational expression to its lowest terms:
* Factorize each of the two polynomials p(x) and q(x)
* Find highest common divisor of p(x) and q(x)
* If h.c.f. = 1 , then the given rational expression [(p(x))/(q(x))] is in its lowest terms
* If h.c.f [!=] 1, then divide the numerator p(x) and denominator q(x) by the h.c.f of p(x) and q(x)
* The rational expression obtained in Step III or Step IV is in its lowest terms
Example: Reduce the rational expression [(x^2-5x-6)/(x^2+3x+2)] to its lowest terms
Solution: Let p(x) = x2 -5x-6 = (x-6)(x+1)
Let q(x) = x2 + 3x +2 = (x+2)(x+1)
Clearly h.c.f of p(x), q(x) = (x+1)

[(x^2-5x-6)/(x^2+3x+2) = [(x-6)(x+1)]/[(x+2)(x+1)]]
[= (x-6)/(x+2)] (cancel out h.c.f = (x+1))

Operations on rational expressions
If [(p(x))/(q(x))] and [(r(x))/(s(x))] are two rational expressions, then their addition is given as:
[(p(x))/(q(x))+(r(x))/(s(x)) = [p(x)s(x)+r(x)q(x)]/(q(x)s(x))] where q(x) [!= ] 0, s(x) [!=] 0
Subtracting rational expressions:
If [(p(x))/(q(x))] and [(r(x))/(s(x))] are two Rational Expressions, then their subtraction is given as:
[(p(x))/(q(x)) - (r(x))/(s(x)) = [p(x) s(x) - r(x) q(x)]/ (q(x) s(x))] where q(x) [!= ] 0, s(x) [!=] 0
Multiplying rational expressions:
If [(p(x))/(q(x))] and [(r(x))/(s(x))] are rational expression then their multiplication is defined as,
["(p(x))/(q(x))* (r(x))/(s(x)) = [p(x) ] where q(x) [!= ] 0, s(x) [!=] 0
Dividing rational expressions:
If [(p(x))/(q(x))] and [(r(x))/(s(x))] are Rational Expressions then their division is defined as,
["[(p(x))/(q(x))]/[(r(x))/(s(x))] = [p(x) * ] where q(x) [!= ] 0, s(x) [!=] 0 and r(x) [!= ] 0,
Solving rational expressions
Below are thye examples on solving rational expressions -
Example 1: If P = [(x+1)/(x-1)] and Q = [(x-1)/(x+1)] then find P + Q
Solution: P + Q = [(x+1)/(x-1) + (x-1)/(x+1)]
= [[(x+1)(x+1) +(x-1)(x-1)]/((x+1)(x-1))]
= [[(x+1)^2 + (x-1)^2]/((x-1)(x+1))] = [(x^2+2x+1+x^2 -2x+1)/((x+1)(x-1))]

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Rational Expressions

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