Identities :- [tex]\sf\purple{{(x+y)^2={\underline{\underline{\pmb{x^{2}+2xy+y^{2}}}}}}}[/tex] [tex]\sf\purple{{(x+y)(x-y)={\underline{\underline{\pmb{x^{2}-y^{2}}}}}}}[/tex] Solution :- [tex] \sf :\implies \red{\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 1}\\\\[/tex] [tex] \sf \: = \: \: \dfrac{ {x}^{8} + 1 + {x}^{4} }{ {x}^{4} } \\\\[/tex] [tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{8} + 2{x}^{4} + 1 – {x }^{4} \bigg)\\\\ [/tex] [tex] \: \: \: \: \: \: \: \: \: \: \: \green{ \sf \: [On \: adding \: and \: subtracting \: {x}^{4} }]\\\\[/tex] [tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {( {x}^{4} + 1) }^{2} – { ({x}^{2}) }^{2} \bigg)\\ [/tex] [tex] \: \: \: \: \: [/tex] [tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg) \bigg( {x}^{4} + 1 + {x}^{2} \bigg)\\[/tex] [tex] \: \: \: \: \: \: \: \: \: [/tex] [tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg) \bigg( {x}^{4} + 1 + {2x}^{2} – {x}^{2} \bigg) \\\\[/tex] [tex] \: \: \: \: \: \: \: \: \: \: \: \green{ \sf \: [On \: adding \: and \: subtracting \: {x}^{2} ]}\\\\\\[/tex] [tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg)\bigg( {( {x}^{2} + 1) }^{2} – {(x)}^{2} \bigg) \\\\[/tex] [tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg) \bigg(( {x}^{2} + 1 + x)( {x}^{2} + 1 – x) \bigg)\\\\\\ [/tex] [tex] \sf \: = \: \: \bigg( \dfrac{ {x}^{4} + 1 – {x}^{2} }{ {x}^{2} } \bigg) \bigg( \dfrac{ {x}^{2} + x + 1 }{x} \bigg) \bigg( \dfrac{ {x}^{2} – x + 1}{x} \bigg)\\\\\\ [/tex] [tex] \sf\red{ \: = \:\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } – 1 \bigg) \bigg( x + \dfrac{1}{x} + 1 \bigg) \bigg(x + \dfrac{1}{x} – 1 \bigg)} \\\\\\[/tex] [tex]\sf\red {Hence }[/tex] [tex] \sf :\implies \purple{\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 1} = \sf \purple \:\purple {\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } – 1 \bigg) \bigg( x + \dfrac{1}{x} + 1 \bigg) \bigg(x + \dfrac{1}{x} – 1 \bigg)}\\\\\\[/tex] ⠀⠀⠀ ━━━━━━━━━━━━━━━━━━━⠀ Reply
Identities :-
Solution :-
[tex] \sf :\implies \red{\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 1}\\\\[/tex]
[tex] \sf \: = \: \: \dfrac{ {x}^{8} + 1 + {x}^{4} }{ {x}^{4} } \\\\[/tex]
[tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{8} + 2{x}^{4} + 1 – {x }^{4} \bigg)\\\\ [/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \green{ \sf \: [On \: adding \: and \: subtracting \: {x}^{4} }]\\\\[/tex]
[tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {( {x}^{4} + 1) }^{2} – { ({x}^{2}) }^{2} \bigg)\\ [/tex]
[tex] \: \: \: \: \: [/tex]
[tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg) \bigg( {x}^{4} + 1 + {x}^{2} \bigg)\\[/tex]
[tex] \: \: \: \: \: \: \: \: \: [/tex]
[tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg) \bigg( {x}^{4} + 1 + {2x}^{2} – {x}^{2} \bigg) \\\\[/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \green{ \sf \: [On \: adding \: and \: subtracting \: {x}^{2} ]}\\\\\\[/tex]
[tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg)\bigg( {( {x}^{2} + 1) }^{2} – {(x)}^{2} \bigg) \\\\[/tex]
[tex] \sf \: = \: \: \dfrac{1}{ {x}^{4} } \bigg( {x}^{4} + 1 – {x}^{2} \bigg) \bigg(( {x}^{2} + 1 + x)( {x}^{2} + 1 – x) \bigg)\\\\\\ [/tex]
[tex] \sf \: = \: \: \bigg( \dfrac{ {x}^{4} + 1 – {x}^{2} }{ {x}^{2} } \bigg) \bigg( \dfrac{ {x}^{2} + x + 1 }{x} \bigg) \bigg( \dfrac{ {x}^{2} – x + 1}{x} \bigg)\\\\\\ [/tex]
[tex] \sf\red{ \: = \:\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } – 1 \bigg) \bigg( x + \dfrac{1}{x} + 1 \bigg) \bigg(x + \dfrac{1}{x} – 1 \bigg)} \\\\\\[/tex]
[tex] \sf :\implies \purple{\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 1} = \sf \purple \:\purple {\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } – 1 \bigg) \bigg( x + \dfrac{1}{x} + 1 \bigg) \bigg(x + \dfrac{1}{x} – 1 \bigg)}\\\\\\[/tex]
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