question_answer 7)

                Factorize:                 (i) \[{{a}^{4}}-{{b}^{4}}\]                               (ii) \[{{p}^{4}}-81\]                          (iii) \[{{x}^{4}}-{{(y+z)}^{4}}\]     (iv) \[{{x}^{4}}-{{(x-z)}^{4}}\] (v) \[{{a}^{4}}-2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\]

Answer:

                (i) \[{{a}^{4}}-{{b}^{4}}\]                                               \[{{a}^{2}}-{{b}^{4}}={{({{a}^{2}})}^{2}}-{{({{b}^{2}})}^{2}}\] \[=({{a}^{2}}-{{b}^{2}})\,({{a}^{2}}+{{b}^{2}})\] \[=(a-b)\,(a+b)({{a}^{2}}+{{b}^{2}})\]     |Using Identity III (ii) \[{{p}^{4}}-81\] \[{{p}^{4}}-81\]\[={{({{p}^{2}})}^{2}}-{{(9)}^{2}}\] \[=({{p}^{2}}-9)\,({{p}^{2}}+9)\]                                |Using Identity III \[=\{{{(p)}^{2}}-{{(3)}^{2}}\}\,\,({{p}^{2}}+9)\] \[=(p-3)\,(p+3)\,({{p}^{2}}+9)\] |Using Identity III (iii) \[{{x}^{4}}-{{(y+z)}^{4}}\] \[{{x}^{4}}-{{(y+z)}^{4}}\]            \[={{({{x}^{2}})}^{2}}-{{\{{{(y+z)}^{2}}\}}^{2}}\] \[=\{{{x}^{2}}-{{(y+z)}^{2}}\}\,\{{{x}^{2}}+{{(y+z)}^{2}}\}\]          |Using Identity III \[=\{x-(y+z)\}\,\{x+(y+z)\}\] \[\{{{x}^{2}}+{{(y+z)}^{2}}\}\]                                    |Using Identity III \[=(x-y-z)\,(x+y+z)\] \[\{{{x}^{2}}+{{(y+z)}^{2}}\}\] (iv) \[{{x}^{4}}-{{(x-z)}^{4}}\] \[{{x}^{4}}-{{(x-z)}^{4}}\] \[={{({{x}^{2}})}^{2}}-{{\{{{(x-z)}^{2}}\}}^{2}}\] \[=\{{{x}^{2}}-{{(x-z)}^{2}}\}\,\{{{x}^{2}}+{{(x-z)}^{2}}\}\]            |Using Identity III \[=\{x-(x-z)\,\}\,\{x+(x-z)\}\]     \[\{{{x}^{2}}+{{(y-z)}^{2}}\}\]                                     |Applying Identity III \[=(x-x+z)\,(x+x-z)\]\[\{{{x}^{2}}+{{(x-z)}^{2}}\}\]                 \[=z(2x-z)\,\,\{{{x}^{2}}+{{(x-z)}^{2}}\}\] \[=z(2x-z)\,({{x}^{2}}+{{x}^{2}}-2xz+{{z}^{2}})\] |Using Identity II \[=z\,(2x-z)\,(2{{x}^{2}}-2xz+{{z}^{2}})\] (v) \[{{a}^{4}}-2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\] \[{{a}^{4}}-2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\] \[={{({{a}^{2}})}^{2}}-2({{a}^{2}})\,({{b}^{2}})\,+{{({{b}^{2}})}^{2}}\] \[={{({{a}^{2}}-{{b}^{2}})}^{2}}\]                                               |Using Identity II \[={{\{(a-b)\,(a+b)\}}^{2}}\]                       |Using Identity III \[={{(a-b)}^{2}}\,{{(a+b)}^{2}}\].