First slide

Binomial theorem for positive integral Index

Question

The expansion of ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}$ is a polynomial of degree

Moderate

A

5
B

6
C

7
D

8

Solution

Using $(a + b)^{5} + (a - b)^{5}$
$= 2 [a^{5} +^{5} C_{2} a^{3} b^{2} +^{5} C_{4} a b^{4}],$
we get ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}$
$= 2 [x^{5} + 10 x^{3} (x^{3} - 1) + 5 x {(x^{3} - 1)}^{2}]$
which is a polynomial of degree $7$ .

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