8th Class Mathematics Sample Paper Mathematics Sample Paper - 10

  • 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       \[3{{x}^{2}}-75=3({{x}^{2}}-25)\]
    \[=3\left[ {{\left( x \right)}^{2}}-{{\left( 5 \right)}^{2}} \right]\]
    \[=3\left( x+5 \right)\left( x-5 \right)\]
    \[\therefore \]      \[\frac{3{{x}^{4}}-1875}{3{{x}^{2}}-75}\]
    \[=\frac{3({{x}^{2}}+25)(x+5)(x-5)}{3(x+5)(x-5)}=({{x}^{2}}+25)\]


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