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) \[\left( 3{{x}^{4}}-1875 \right)\div \left( 3{{x}^{2}}-75 \right)\]

     

Answer:

(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       \[3{{x}^{2}}-75=3({{x}^{2}}-25)\]

\[=3\left[ {{\left( x \right)}^{2}}-{{\left( 5 \right)}^{2}} \right]\]

\[=3\left( x+5 \right)\left( x-5 \right)\]

\[\therefore \]      \[\frac{3{{x}^{4}}-1875}{3{{x}^{2}}-75}\]

\[=\frac{3({{x}^{2}}+25)(x+5)(x-5)}{3(x+5)(x-5)}=({{x}^{2}}+25)\]