Relationship between LCM and GCD :
f(x) ⋅ g(x) = LCM (f(x), g(x)) ⋅ GCD (f(x), g(x))
LCM(f(x), g(x)) = [f(x) ⋅ g(x)]/GCD (f(x), g(x))
Find the LCM of each pair of the following polynomials
(i) x2-5x+6, x2+4x-12 whose GCD is (x-2)
(ii) x4+3x3+6x2+5x+3, x4+2x2+x+2 whose GCD is x2+x+1
(iii) 2x3+15x2+2x-35, x4+8x2+4x-21 whose GCD is x+7
(iv) 2x3-3x2-9x+5, 2x4-x3-10x2-11x+8 whose GCD is 2x-1
(i) Answer :
LCM ⋅ GCD = f(x) ⋅ g(x)
LCM = [f(x) ⋅ g(x)]/GCD
f(x) = x2-5x+6
g(x) = x2+4x-12
GCD = (x-2)
x2-5x+6 = (x-2)(x-3)
x2+4x-12 = (x+6)(x-2)
LCM = (x-2)(x-3)(x+6)(x-2)/(x-2)
By canceling common factors, we get
LCM = (x-2) (x-3) (x+6)
So, the required LCM is (x-2) (x-3) (x+6).
(ii) Answer :
x4+3x3+6x2+5x+3, x4+2x2+x+2 whose GCD is x2+x+1
LCM = [f(x) ⋅ g(x)]/GCD
f(x) = x4+3x3+6x2+5x+3
g(x) = x4+2x2+x+2
GCD = x2+x+1
LCM = [(x4+3x3+6x2+5x+3) (x4+2x2+x+2) ]/(x2+x+1)
To simplify this, we have to use long division.
LCM = (x2+2x+3) (x4+2x2+x+2)
So, the required LCM is (x2+2x+3) (x4+2x2+x+2).
(iii) Answer :
2x3+15x2+2x-35, x4+8x2+4x-21 whose GCD is x+7
LCM = [f(x) ⋅ g(x)]/GCD
f(x) = 2x3+15x2+2x-35
g(x) = x4+8x2+4x-21
GCD = x+7
LCM = [(2x3+15x2+2x-35) (x4+8x2+4x-21)]/(x + 7)
To simplify this we have to use long division.
LCM = (2x2+x-5) (x4+8x2+4x-21)
So, the required LCM is (2x2+x-5) (x4+8x2+4x-21).
(iv) Answer :
2x3-3x2-9x+5, 2x4-x3-10x2-11x+8 whose GCD is 2x-1
LCM = [f(x) ⋅ g(x)]/GCD
f(x) = 2x3-3x2-9x+5
g(x) = 2x4-x3-10x2-11x+8
GCD = 2x-1
LCM = [(2x3-3x2-9x+5) (2x4-x3-10x2-11x+8)]/(2x-1)
To simplify this we have to use long division.
LCM = (x3-5x-8) (2x3-3x2-9x+5)
So, the LCM is (x3-5x-8) (2x3-3x2-9x+5).
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