Question 6: The HCF and LCM of two polynomials is \((x+a)\) and \(12x^2(x+a)(x^2-a^2)\) respectively. Find the product of the two polynomials.
Note: The question given in the book is incorrect. The correct form of the question should be: The HCF and LCM of two polynomials is \((x+a)\) and \(12x^2(x-a)(x^2+a^2)\) respectively. Find the product of the two polynomials.
Solution:
Since we know that $\displaystyle p( x) \times q( x) =HCF\times LCM$
Let $\displaystyle p( x) \times q( x)$ is our required product of polynomials
$\displaystyle p( x) \times q( x) =( x+a) \times 12x^{2} (x-a)(x^{2} +a^{2} )$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =12x^{2}( x+a)( x-a)\left( x^{2} +a^{2}\right)$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =12x^{2}\left( x^{2} -a^{2}\right)\left( x^{2} +a^{2}\right)$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =12x^{2}\left(\left( x^{2}\right)^{2} -\left( a^{2}\right)^{2}\right)$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =12x^{2}\left( x^{4} -a^{4}\right)$