Question 5: The LCM and HCF of two polynomials \(p(x)\) and \(q(x)\) are \(36x^3(x+a)(x^3-a^3)\) and \(x^2(x-a)\) respectively. If \(p(x)=4x^2(x^2-a^2)\), find \(q(x)\).
Solution:
Since we know that $\displaystyle p( x) \times q( x) =HCF\times LCM$
$\displaystyle 4x^{2} (x^{2} -a^{2} )\times q( x)$$\ =\ $$x^{2} (x-a)\times 36x^{3} (x+a)(x^{3} -a^{3} )$
$\displaystyle q( x) =\frac{x^{2} (x-a)\times 36x^{3} (x+a)(x^{3} -a^{3} )}{4x^{2} (x^{2} -a^{2} )}$
$\displaystyle \ \ \ \ \ \ \ =\frac{(x-a)\times 9x^{3} (x+a)(x^{3} -a^{3} )}{(x -a )( x+a)}$
$\displaystyle \ \ \ \ \ \ \ =9x^{3} (x^{3} -a^{3} )$