LEAST COMMON MULTIPLE (LCM) OF POLYNOMIALS
L.C.M. OF POLYNOMIALS The least common multiple (L.C.M.) of two or more polynomials is the polynomial of the lowest degree, having smallest numerical coefficient which is exactly divisible by the given polynomials and whose coefficient of highest degree term has the same sign as the sign of the coefficient of highest degree term in their product.
In order to find the LCM of two or more polynomials, we may use the following algorithm.
ALGORITHM
STEP I Resolve each of the given polynomials into factors and express them as a product of powers of irreductible factors.
STEP II List all the irreducible factors (once only) occurring in the given polynomials. For each of these factors,find the greatest exponent in the factorized form of the given polynomials.
STEP III Raise each irreducible factor to the greatest exponent found in step II and multiply them to get the LCM.
Find the LCM of the following polynomials:
SOLUTION We have,
\(f(x) = 7x^3+2x^2-16x-32\) \(f(2) 7×2^3+2×2^2-16x^2-32 = 0\) \(x – 2 is a factor of f(x)\)Now,
Since -1 is a factor of constant terming (x) suchthatg (-l) = —1+6-11+6 = 0
\((x + 1) is a factor of g(x)\)Now,
We observe that the irreducible divisors of f(x) and g (x) are \((x — 2), (7x^2 + 16x + 16), (x + 1), (x + 2) and (x + 3).\) The highest exponent of each of these divisors is 1.
RequiredLCM =\((x+1)(x+2)(x-2)(x+3)(x^2+16x+16).\)