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

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7.Binomial Theorem

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

$5$

$6$

$7$

$8$

Solution

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

The binomial expression is $=2\left[C_{0}^{5} x^{5}+C_{2}^{5} x^{3}\left(\sqrt{1-x^{3}}\right)^{2}+C_{4}^{5} x\left(1-x^{3}\right)^{\frac{3}{2}}\right]$

$=2\left\{C_{0}^{5} x^{5}+C_{2}^{5} x^{3}\left(1-x^{3}\right)+C_{4}^{5} x\left[1-\frac{3}{2} x^{3}+\frac{3}{2} \times \frac{1}{6} x^{6}\right]\right\}$

$=2\left\{C_{0}^{5} x^{5}+C_{2}^{5}\left(x^{3}-x^{6}\right)+C_{4}^{5}\left(x-\frac{3}{2} x^{4}+\frac{3}{4} x^{7}\right)\right\}$

Hence degree of the polynomial is 7

Standard 11

Mathematics

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