The sum of the coefficients of all odd degree | vedclass

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(C) Let $f(x) = (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5$.Using the binomial expansion formula $(a+b)^n + (a-b)^n = 2[\binom{n}{0}a^n + \binom{

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(C) Let $f(x) = (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5$.Using the binomial expansion formula $(a+b)^n + (a-b)^n = 2[\binom{n}{0}a^n + \binom{n}{2}a^{n-2}b^2 + \binom{n}{4}a^{n-4}b^4 + \dots]$,we get:$f(x) = 2[\binom{5}{0}x^5 + \binom{5}{2}x^3(\sqrt{x^3-1})^2 + \binom{5}{4}x(\sqrt{x^3-1})^4]$$f(x) = 2[x^5 + 10x^3(x^3-1) + 5x(x^3-1)^2]$$f(x) = 2[x^5 + 10x^6 - 10x^3 + 5x(x^6 - 2x^3 + 1)]$$f(x) = 2[x^5 + 10x^6 - 10x^3 + 5x^7 - 10x^4 + 5x]$$f(x) = 10x^7 + 20x^6 + 2x^5 - 20x^4 - 20x^3 + 10x$.The odd degree terms are $10x^7$,$2x^5$,$-20x^3$,and $10x$.The sum of the coefficients of these terms is $10 + 2 - 20 + 10 = 2$.

The sum of the coefficients of all odd degree terms in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5$,where $x > 1$,is:

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