• # question_answer Factorise and divide the following: (a) $\left( 2{{x}^{3}}-12{{x}^{2}}+16x \right)\div \left( x-2 \right)\left( x-4 \right)$ (b) $\left( 3{{x}^{4}}-1875 \right)\div \left( 3{{x}^{2}}-75 \right)$

 (a) $\left( 2{{x}^{3}}-12{{x}^{2}}+16x \right)\div \left( x-2 \right)\left( x-4 \right)$ $\because$ $2{{x}^{3}}-12{{x}^{2}}+16x=2x\left( {{x}^{2}}-6x+8 \right)$ $=2x\text{ }\left( {{x}^{2}}-4x-2x+8 \right)$ $=2x\left[ x\left( x-4 \right)-2\left( x-4 \right) \right]$ $=2x\left[ \left( x-4 \right)\left( x-2 \right) \right]$ $=2x\left( x-2 \right)\left( x-4 \right)$ $\therefore$   $\frac{2{{x}^{2}}-12{{x}^{2}}+16x}{(x-2)(x-4)}=\frac{2x(x-2(x-4)}{(x-2)(x-4)}=2x$ (b) $(3{{x}^{4}}-1875)\div (3{{x}^{2}}-75)$ $\because$       $3{{x}^{4}}-1875=3({{x}^{4}}-625)$ $=3[{{({{x}^{2}})}^{2}}-{{(25)}^{2}}]$ $=3[({{x}^{2}}+25)({{x}^{2}}-25)]$ $=3[({{x}^{2}}+25)({{x}^{2}}-{{5}^{2}})]$ $=3[({{x}^{2}}+25)(x+5)(x-5)]$ and       $3{{x}^{2}}-75=3({{x}^{2}}-25)$ $=3\left[ {{\left( x \right)}^{2}}-{{\left( 5 \right)}^{2}} \right]$ $=3\left( x+5 \right)\left( x-5 \right)$ $\therefore$      $\frac{3{{x}^{4}}-1875}{3{{x}^{2}}-75}$ $=\frac{3({{x}^{2}}+25)(x+5)(x-5)}{3(x+5)(x-5)}=({{x}^{2}}+25)$