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Rational Expressions in Lowest terms
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Rational Expressions in Lowest terms
STEP I Factorise each of the two polynomials p(x) and q(x).
STEP II Find g.c.d. of  p(x) and q(x)
STEP III If g.c.d. of  p(x) and q(x)is one, then the given rational expression p(x)/q(x) is in 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 IV The rational esxpression obtained in step III is in its lowest terms.
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Example: Reduce to its lowest terms.
Solution: You know that a + b + c = 0 a3 + b3 + c3 = 3abc
(x2 - y2) + (y2 - z2) + (z2 - x2) = 0
(x2 - y2)3 + (y2 - z2)3 + (z2 - x2)3 = 3(x2 - y2)(y2 - z2)(z2 - x2)
= 3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)
= 3(x - y)(y - z)(z - x)(x + y)(y + z)(z + x)
Also,
(x - y) + (y - z) + (z - x) = 0
(x - y)3+ (y - z)3+ (z - x)3 = 3(x - y)(y - z)(z - x)
   
    = (x + y)(y + z)(z + x)
 
   
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