The expression [x + (x^3-1)^{1/2}]^5 + [x - ( | vedclass

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Question

The given expression is $\left[x+\left(x^{3}-1\right)^{\frac{1}{3}}\right]^{\frac{5}{3}}+\left[x-\left(x^{3}-1\right)^{\frac{1}{2}}\right]$

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Vedclass · Accepted Answer

$7$

Vedclass · Answer

The given expression is $\left[x+\left(x^{3}-1ight)^{\frac{1}{3}}ight]^{\frac{5}{3}}+\left[x-\left(x^{3}-1ight)^{\frac{1}{2}}ight]$

The binomial expression is $=2\left[C_{0}^{5} x^{5}+C_{2}^{5} x^{3}\left(\sqrt{1-x^{3}}ight)^{2}+C_{4}^{5} x\left(1-x^{3}ight)^{\frac{3}{2}}ight]$

$=2\left\{C_{0}^{5} x^{5}+C_{2}^{5} x^{3}\left(1-x^{3}ight)+C_{4}^{5} x\left[1-\frac{3}{2} x^{3}+\frac{3}{2} 	imes \frac{1}{6} x^{6}ight]ight\}$

$=2\left\{C_{0}^{5} x^{5}+C_{2}^{5}\left(x^{3}-x^{6}ight)+C_{4}^{5}\left(x-\frac{3}{2} x^{4}+\frac{3}{4} x^{7}ight)ight\}$

Hence degree of the polynomial is 7

The expression $[x + (x^3-1)^{1/2}]^5 + [x - (x^3-1)^{1/2}]^5$ is a polynomial of degree :

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