FIND LCM WHEN POLYNOMIALS AND GCD ARE GIVEN

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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