FIND OTHER POLYNOMIAL WHEN ONE POLYNOMAILS ITS LCM AND GCD ARE GIVEN

Example :

Find the other polynomial q(x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively.

(i)  (x+1)² (x+2)², (x+1) (x+2), (x+1)² (x+2)

(ii)  (4x+5)³ (3x-7)³, (4x+5) (3x-7)², (4x+5)³ (3x-7)²

(iii)  (x⁴-y⁴)(x⁴+x²y²+y⁴), x²-y², x⁴-y⁴

(iv)  (x³-4x) (5x+1), (5x²+x), (5x³-9x²-2x)

(vi)  2(x+1) (x²-4), (x+1), (x+1) (x-2)

(v)  (x-1) (x-2) (x²-3x+3), (x-1), (x³-4x²+6x-3)

(vi)  2(x+1) (x²-4), (x+1), (x+1) (x-2)

(i)  Answer :

(x+1)² (x+2)², (x+1) (x+2), (x+1)² (x+2)

LCM ⋅ GCD  =  p(x) ⋅ q(x)

L.C.M  =  (x+1)² (x+2)²

GCD  =  (x+1) (x+2)

p(x)  =  (x+1)² (x+2)

q(x)  =  [LCM ⋅ GCD]/p(x)

q(x)  =  [(x+1)²(x+2)² (x+1) (x+2)]/(x+1)² (x+2)

q(x)  =  (x+2)² (x+1)

So, the other polynomial is (x+2)² (x+1).

(ii)  Answer :

(4x+5)³ (3x-7)³, (4x+5) (3x-7)², (4x+5)³ (3x-7)²

LCM ⋅ GCD  =  p(x) ⋅ q(x)

LCM  =  (4x+5)³ (3x-7)³

GCD  =  (4x+5) (3x-7)²

p(x)  =  (4x+5)³ (3x-7)²

q(x)  =  [LCM ⋅ GCD]/p(x)

q(x)  =  [(4x+5)³ (3x-7)³(4x+5) (3x-7)²]/(4x+5)³ (3x-7)²

q(x)  =  (3x-7)³ (4x+5)

(iii)  Answer :

(x⁴-y⁴)(x⁴+x²y²+y⁴), x²-y², x⁴-y⁴

LCM ⋅ GCD  =  p(x) ⋅ q(x)

LCM  =  (x⁴-y⁴)(x⁴+x²y²+y⁴)

GCD  =  x²-y²

p(x)  =  x⁴-y⁴

q(x)  =  [LCM ⋅ GCD]/p(x)

q(x)  =  [(x⁴-y⁴)(x⁴+x²y²+y⁴)(x²-y²)]/(x⁴-y⁴)

q(x)  =  (x⁴+x²y²+y⁴)(x²-y²)

So, the other polynomial is (x⁴+x²y²+y⁴)(x²-y²).

(iv)  Answer :

(x³-4x) (5x+1), (5x²+x), (5x³-9x²-2x)

LCM ⋅ GCD  =  p(x) ⋅ q(x)

LCM  =  (x³-4x) (5x+1)

GCD  =  5x²+x  =  x(5x+1)

p(x)  =  5x³-9x²-2x ==>  x(5x²-9x-2)

x(5x²-9x-2)  ==> x (5x+1)(x-2)

q(x)  =  [LCM ⋅ GCD]/p(x)

q(x)  =  [(x³-4x) (5x+1)(5x+1)]/[x (5x+1)(x-2)]

q(x)  =  [(x²-4)(5x+1)]/(x-2)

q(x)  =  [(x+2)(x-2)(5x+1)]/(x-2)

q(x)  =  (x+2)(5x+1)

So, the other polynomial is (x+2)(5x+1).

(v)  Answer :

(x-1) (x-2) (x²-3x+3), (x-1), (x³-4x²+6x-3)

LCM ⋅ GCD  =  p(x) ⋅ q(x)

LCM  =  (x-1) (x-2) (x²-3x+3)

GCD  =  (x-1)

p(x)  =  x³-4x²+6x-3

q(x)  =  [LCM ⋅ GCD]/p(x)

         =  [(x-1)(x-2)(x²-3x+3)(x-1)]/(x³-4x²+6x-3)

let us use synthetic division to find factors of the cubic polynomial.

=  [(x-1)(x-2) (x²-3x+3)(x-1)]/(x-1) (x²-3x+3)

=  (x-2)(x-1)

So, the other polynomial is (x-2)(x-1).

(vi)  Answer :

2(x+1) (x²-4), (x+1), (x+1) (x-2)

LCM ⋅ GCD  =  p(x) ⋅ q(x)

LCM  =  2(x+1) (x²-4)

GCD  =  (x+1)

p(x)  =  (x+1) (x-2)

q(x)  =  [LCM ⋅ GCD]/p(x)

q(x)  =  [2(x+1) (x²-4)(x+1)]/(x+1) (x-2)

q(x)  =  [2(x+1) (x+2) (x-2) (x+1)]/(x+1) (x-2)

q(x)  =  2(x+1) (x+2)

So, the other polynomial is 2(x+1) (x+2).

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