HOMEWORK

Greatest common divisor (g.c.d) or hcf of two polynomials are highest degree common divisor of both polynomials. Is the product of the common factors of both polynomials to the less exponent

Example:      G.C.D of polynomials  P(x) = 2x2-x-1   and   Q(x) = 4x2+8x+3:
                           
                            P(x)=2x2-x-1=(2x+1)(x-1)
                            Q(x)=4x2+8x+3=(2x+1)(2x+3) 

In both polynomial only (2x+1) is only common divisor with least exponent 1

so GCD of p(x) and q(x) is (2x+1)

GCD = 2x + 1

Least common multiple (l.c.m) of two polynomials are common and uncommon factors raised to the highest exponents:

Example:       L.C.M of Polynomials  P(x) =18x4-36x3+18x2  and  Q(x) = 45x6-45x3
                           
                        P(x)=2.32x2(x-1)2

                         Q(x)=325x3(x-1)(x2+x+1)

In both polynomials irreducible factors are 2, 3, 5, x, (x-1) and (x2+x+1). 

So LCM =213251x3(x-1)2(x2+x+1)

LCM =90x3(x-1)2(x2+x+1)