8th Class Mathematics Sample Paper Mathematics Sample Paper - 3

  • question_answer            
    Factorise:
    (a) \[{{a}^{4}}-{{b}^{4}}\]                           
    (b) \[{{p}^{4}}-81\]
    (c) \[{{x}^{4}}-{{(y+2)}^{4}}\]                 
    (d) \[{{x}^{4}}-{{(x-z)}^{4}}\]
    (e) \[{{a}^{4}}-2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\]

    Answer:

    (a) Using   \[{{a}^{2}}-{{b}^{2}}=(a-b)(a+b)\]
    \[{{a}^{4}}-{{b}^{4}}={{({{a}^{2}})}^{2}}-{{({{b}^{2}})}^{2}}\]
    \[=({{a}^{2}}+{{b}^{2}})({{a}^{2}}-{{b}^{2}})\]
    \[=({{a}^{2}}+{{b}^{2}})(a+b)(a-b)\]                                                                                                                                  
    (b)   \[{{p}^{4}}-81={{\left( {{p}^{2}} \right)}^{2}}-{{\left( 9 \right)}^{2}}\]
    \[=({{p}^{2}}+9)\left( {{p}^{2}}-9 \right)\] \[[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)]~\]
    \[=\left( {{p}^{2}}+9 \right)\left( p-3 \right)\left( p+3 \right)~~\]                                                                                                              
    (c)   \[{{x}^{4}}-{{(y+2)}^{4}}={{({{x}^{2}})}^{2}}-{{[{{(y+2)}^{2}}]}^{2}}\]
    \[=[({{x}^{2}})+{{(y+2)}^{2}}]\,\,[({{x}^{2}})-{{(y+2)}^{2}}]\]
    \[=[{{(x)}^{2}}+{{(y+2)}^{2}}][(x-y-z)(x+y+2)]\]                                                                                                     
    (d)   \[{{x}^{4}}-{{(x-z)}^{4}}={{({{x}^{2}})}^{2}}-{{[{{(x-z)}^{2}}]}^{2}}\]
    \[=[{{x}^{2}}-{{(x-z)}^{2}}]\,[{{x}^{2}}+{{(x-z)}^{2}}]\]
    \[=\left[ \left( x-x+z \right)\left( x+x-\text{ }z \right) \right]\text{ }\left[ ({{x}^{2}}+{{(x-z)}^{2}} \right]~\]                                                                                                   
    \[=\text{ }z\left( 2x-z \right)\text{ }\left[ {{x}^{2}}+{{\left( x \right)}^{2}}+{{\left( z \right)}^{2}}-2xz \right]\]
    \[=z(2x-z)[2{{x}^{2}}-2xz+{{z}^{2}}]\]
    (e)    \[{{a}^{4}}-2{{a}^{2}}{{\text{b}}^{2}}+{{b}^{4}}={{\left( {{a}^{2}} \right)}^{2}}+{{\left( {{b}^{2}} \right)}^{2}}-2\left( {{a}^{2}} \right)\left( {{b}^{2}} \right)\]
    \[={{({{a}^{2}}-{{b}^{2}})}^{2}}\]
    \[=[({{a}^{2}}-{{b}^{2}})({{a}^{2}}+{{b}^{2}})]\]
    \[=[(a-b)(a+b)({{a}^{2}}+{{b}^{2}})]\]        


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