Question 65

Sum of the factors of $$4b^2c^2 - (b^2 + c^2 - a^2)^2$$ is

Solution

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

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

Using, $$x^2-y^2=(x-y)(x+y)$$, where $$x=2bc$$ and $$y=b^2+c^2-a^2$$

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

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

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

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

Thus, sum of factors = $$(a-b+c)+(a+b-c)+(b+c-a)+(b+c+a)$$

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

=> Ans - (B)

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