Question

9. Factorize:
${x}^{4} + {x }^{2} + 25$

fastest answer with explanation is the brainliest

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Answer 1

$\huge\mathbb{SOLUTION:-}$

$\sf \bullet x{}^{4}+x{}^{2}+25$

We have to factorize this

Now :-

• 2 identities are used

$\boxed{\sf{(a+b){}^{2}=a{}^{2}+b{}^{2}+2ab}}$

$\boxed{\sf{a{}^{2}-b{}^{2}=(a+b)(a-b)}}$

$\sf\underline{\pink{\:\:\:\: Solution:-\:\:\:\:}}$

$\sf \implies x{}^{4}+x{}^{2}+25\\ \\ \sf\implies (x{}^{4}+10x{}^{2}+25)-9x{}^{2}\\ \\ \sf\implies it\: became\:in\:the\:form\:of\\ \\ \sf\:\:\:a{}^{2}-b{}^{2}=(a+b)(a-b)\\ \\ \sf\implies (x{}^{2}+5){}^{2}-(3x){}^{2}\\ \\ \sf\implies (x{}^{2}+5+3x)(x{}^{2}+5-3x)\\ \\ \sf\implies \:or\: (x{}^{2}+3x+5)(x{}^{2}-3x+5)$

$\boxed{\sf{\therefore x{}^{4}+x{}^{2}+25=(x{}^{2}+3x+5)(x{}^{2}-3x+5)}}$

Answer 2

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: SOLUTION \: \: } \mid}}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \mathcal{ \underline{ \underline{ \blue{ \: \: TO \: \: \: FACTORISE : \leadsto \: \: }}}}$

$\: \: \: \: \: \tt{x {}^{4} + x {}^{2} + 25 } \\ \\ \tt{ = x {}^{4} + 10x {}^{2} - 9x {}^{2} + 25} \\ \\ \tt{ = (x {}^{4} + 10x {}^{2} + 25) - 9x {}^{2} }$

$\underline{ \mathfrak{Now,}} \\ \\ \: \: \: \underline{\text{ \: Factorizing the expression \: } } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \bold{x {}^{4} + 10x {}^{2} + 25}}}$

$\underline{ \text{ \: \: Identity \: \: used :- \: \: }} \\ \\ \: \: \: \: \: \: \underline{ \boxed{ \bold{ \: \: a {}^{2} + 2ab + b {}^{2} = (a + b) {}^{2} }}}$

$\: \: \: \: \: \: \: \: \: \: \tt{x {}^{4} + 10x {}^{2} + 25 } \\ \\ \tt{ = (x {}^{2} ) {}^{2} + 2 \times x {}^{2} \times 5 + (5) {}^{2} } \\ \\ \tt{ = (x {}^{2} + 5) {}^{2} } \:$

$\tt{ \therefore \: \underline{\red{ \: (x {}^{4} + 10x {}^{2} + 25) - 9x {}^{2} = (x {}^{2} + 5) {}^{2} - 9x {}^{2} \: } } }$

$\underline{ \mathfrak{Now,}} \\ \\ \: \: \: \underline{\text{ \: Factorizing the expression \: } } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \bold{(x {}^{2} + 5) {}^{2} - 9x {}^{2} }}}$

$\underline{ \text{ \: \: Identity \: \: used :- \: \: }} \\ \\ \: \: \: \: \: \: \underline{ \boxed{ \bold{ \: \: a {}^{2} - b {}^{2} = (a + b) (a - b)}}}$

$\: \: \: \: \: \: \: \tt{(x {}^{2} + 5) {}^{2} - 9x {}^{2} } \\ \\ \tt{ = [(x {}^{2} + 5) + 9x] [x {}^{2} + 5) - 9x ] } \\ \\ \tt{ = (x {}^{2} + 5 + 9x)(x {}^{2} + 5 - 9x)} \\ \\ \tt{ = (x {}^{2} + 9x + 5)(x {}^{2} - 9x + 5)}$

$\therefore \: \boxed{ \boxed{ \tt {\red{ \: x {}^{4} + x {}^{2} + 25 = (x {}^{2} + 9x + 5)(x {}^{2} - 9x + 5) \: }}}}$