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:-
$lcm = (x - 1)(x - 2)( {x}^{2} - 3x + 3) = gcd = (x - 1) \\ q(x) = ( {x}^{ 3} - 4 {x}^{2} + 6x - 3) \\ = ( {x}^{3} - {x}^{2} - 3 {x}^{2} + 3x + 3x - 3) \\ q(x) = {x}^{2} (x - 1) - 3x(x - 1) + 3(x - 1) \\ q(x) = (x - 1)( {x}^{2} - 3x + 3) \\ here \: \\ gcd × lcm = q(x) × p(x) \\ (x - 1) × (x - 1)(x - 2)( {x}^{2} - 3x + 3) = (x - 1)( {x}^{2} - 3x + 3) × p(x) \\ p(x) = (x - 1)(x - 2) \: ans$
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