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Algebra I: Largest Common Factor
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Largest Common Factor: Example

Example: Find the L.C.M. of the following expressions:
    p2 - q2 - r2 + 2qr, (p + q - r)2, p2 - q2 + r2 + 2pr
Solution: Given,
p2 - q2 - r2 + 2qr  = p2 - (q2 + r2 - 2qr)
 = p2 - (q - r)2
 =[p + (q - r)] [p - (q - r)]
 =(p + q - r)(p - q + r)
(p + q - r)2  = (p + q - r)(p + q - r)
p2 - q2 + r2 + 2pr  = (p2 + r2 + 2pr) - q2
 = (p + r)2 - q2
 = (p + r - q)(p + r + q)
 = (p - q + r)(p + q + r)

Irreducible factors of the given expressions are: p + q - r, p - q + r and p + q + r. Their respective highest exponents are 2,1 and 1.
Therefore,
                   Required L.C.M. = (p - q + r)(p + q + r)(p + q - r)2
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Division of Rational Expressions
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nth term of an A.P.
Right Circular Cone
Linear Equation: Conditions for Consistency
Example: Find the L.C.M. of p(x)= -x2 - x + 12 and q(x)= x2 + 2x - 8.
Solution: Given,
p(x)
 = -x2 - x + 12
 = -(x2 + x - 12)
 = -(x + 4)(x - 3)
and q(x)
 = x2 + 2x - 8
 = (x + 4)(x - 2)
Irreducible factors of the polynomials p(x) and q(x) are (x + 4),(x - 2) and (x - 3). Their respective highest exponents are 1 each.
Therefore,
                 Required L.C.M. =  (x + 4)(x - 2)(x - 3)
 
   
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