Help to factorize!
Factorize the given expression:$2a^{2}b^{2}+2b^{2}c^{2}+2c^{2}a^{2}-a^{4}-b^{4}-c^{4}$.I need help immediately!Please!
Re: Help to factorize!
@Dear Mahi, perhaps you wrote it in haste and did not look that your factorization would make all the terms negative. The actual factors look like
$(a+b+c)(a+b-c)(c+a-b)(c-a+b)$
$(a+b+c)(a+b-c)(c+a-b)(c-a+b)$
Re: Help to factorize!
Give those steps please!@mamun vai.
Re: Help to factorize!
Sorry, deleted the post now, thanks for noting.SMMamun wrote:@Dear Mahi, perhaps you wrote it in haste and did not look that your factorization would make all the terms negative.
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Nur Muhammad Shafiullah | Mahi
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Nur Muhammad Shafiullah | Mahi
Re: Help to factorize!
Dear Abdullah, hope it helps
$2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4$
$=-(a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2)$
$=-\left\{(a^4+b^4+c^4-2a^2b^2+2b^2c^2-2c^2a^2)-4b^2c^2\right\}$
$=-\left\{(a^2-b^2-c^2 )^2-(2bc)^2\right\}$
$=-\left\{a^2-b^2-c^2+2bc\right\}\left\{a^2-b^2-c^2-2bc\right\}$
$=-\left\{a^2-(b^2+c^2-2bc)\right\}\left\{a^2-(b^2+c^2+2bc)\right\}$
$=-\left\{a^2-(b-c)^2\right\}\left\{a^2-(b+c)^2\right\}$
$=-(a+b-c)(a-b+c)(a+b+c)(a-b-c)$
$=(a+b+c)(a+b-c)(c+a-b)(c-a+b)$
$2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4$
$=-(a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2)$
$=-\left\{(a^4+b^4+c^4-2a^2b^2+2b^2c^2-2c^2a^2)-4b^2c^2\right\}$
$=-\left\{(a^2-b^2-c^2 )^2-(2bc)^2\right\}$
$=-\left\{a^2-b^2-c^2+2bc\right\}\left\{a^2-b^2-c^2-2bc\right\}$
$=-\left\{a^2-(b^2+c^2-2bc)\right\}\left\{a^2-(b^2+c^2+2bc)\right\}$
$=-\left\{a^2-(b-c)^2\right\}\left\{a^2-(b+c)^2\right\}$
$=-(a+b-c)(a-b+c)(a+b+c)(a-b-c)$
$=(a+b+c)(a+b-c)(c+a-b)(c-a+b)$