Is it correct that the sum of the factors of the polynomial $4b^2c^2-(b^2+c^2-a^2)^2$ is equal to $2(a+b+c)$?

Given: The polynomial: $4b^2c^2-(b^2+c^2-a^2)^2$.

To do: To check whether the sum of the factors of the polynomial $4b^2c^2-(b^2+c^2-a^2)^2$ is equal to $2(a+b+c)$.

Solution:
 

$4b^2c^2-(b^2+c^2-a^2)^2$

$=( 2bc)^2-( b^2+c^2-a^2)^2$

$=( 2bc+b^2+c^2-a^2)(2bc-b^2-c^2+a^2))$

$=( ( b+c)^2-a^2)( a^2-( b-c)^2)$

$=( b+c-a)(b+c+a)(a-b+c)(a+b-c)$

Therefore, $( b+c-a),\ (b+c+a),\ (a-b+c),\ (a+b-c)$ are the factors of $4b^2c^2-(b^2+c^2-a^2)^2$.

The sum of the factors$=( b+c-a)+(b+c+a)+(a-b+c)+(a+b-c)$

$=b+c-a+b+c+a+a-b+c+a+b-c$

$=2a+2b+2c$

$=2( a+b+c)$

Thus, it is correct that the sum of the factors of the polynomial $4b^2c^2-(b^2+c^2-a^2)^2$ is equal to $2(a+b+c)$.

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Updated on: 10-Oct-2022

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