Question

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

Answer 1

SOLUTION IS IN THE ATTACHMENT

Greatest Common Divisor (GCD) :
Greatest Common Divisor (GCD) or (HCF) of two or more algebraic expressions is the expression of highest degree which divides each of them without remainder.

Least Common Multiple(LCM): The least common multiple of two or more algebraic expressions is the expression of lowest degree which is divisible by each of them without remainder.

RELATION BETWEEN LCM AND GCD :

The product of any two polynomials is equal to the product of their LCM and GCD. f(x) g(x) = LCM (f(x) , g(x)) × GCD (f(x) , g(x)).

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Answer 2

solution:-

given by:-
$lcm = ( {x}^{4} - {y}^{4} )( {x}^{4} + {x}^{2} {y}^{2} + {y}^{4} ) \\ gcd = {x}^{2} - {y}^{2} \\ q(x) = {x}^{4} - {y}^{4} \\ we \: know \: that \: \\ lcm \times gcd = q(x) \times p(x) \\ ( {x}^{4} - {y}^{4} )( {x}^{4} + {x}^{2} {y}^{2} + {y}^{4} ) \times ({x}^{2} - {y}^{2}) =( {x}^{4} - {y}^{4}) \times p(x) \\ p(x) = \frac{ ( {x}^{4} - {y}^{4} )( {x}^{4} + {x}^{2} {y}^{2} + {y}^{4} ) \times ({x}^{2} - {y}^{2})}{( {x}^{4} - {y}^{4})} \\ p(x) = ( {x}^{4} + {x}^{2} {y}^{2} + {y}^{4} ) ({x}^{2} - {y}^{2}) ans$
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