Question 3: Find LCM of the following expressions by using prime factorization method:
\((iv)\) \(\; \)\(x^4-16,x^3-4x \)
Solution:
$\displaystyle x^{4} -16=\left( x^{2}\right)^{2} -4^{2}$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ =\left( x^{2} -4\right)\left( x^{2} +4\right)$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ =\left( x^{2} -2^{2}\right)\left( x^{2} +4\right)$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ =( x-2)( x+2)\left( x^{2} +4\right)$
$\displaystyle x^{3} -4x=x\left( x^{2} -4\right)$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ =x\left( x^{2} -2^{2}\right)$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ =x( x-2)( x+2) \ $
$\displaystyle \text{Common Factors} =( x-2)( x+2)$$\ =\ $$x^{2} -4$
$\displaystyle \text{Non-Common Factors} =x\left( x^{2} +4\right)$
$\displaystyle \text{LCM}$$\ =\ $$\text{Common Factors} \times \text{Non Common Factors}$
$\displaystyle \text{LCM} =\left( x^{2} -4\right) \times x\left( x^{2} +4\right)$
$\displaystyle \text{LCM} =x\left( x^{4} -16\right)$