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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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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