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Binomial Theorem — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsBinomial Theorem5 Marks
Question
Evaluate the following
$\Big(\sqrt{\text{x}+1}+\sqrt{\text{x}-1}\Big)^6+\Big(\sqrt{\text{x}+1}-\sqrt{\text{x}-1}\Big)^6$
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Answer

$\Big(\sqrt{\text{x}+1}+\sqrt{\text{x}-1}\Big)^6+\Big(\sqrt{\text{x}+1}-\sqrt{\text{x}-1}\Big)^6$
$={^6\text{C}}_0\big(\sqrt{\text{x}+1}\big)^6+{^6\text{C}}_1\big(\sqrt{\text{x}+1}\big)^5\big(\sqrt{\text{x}-1}\big)+{^6\text{C}}_2\big(\sqrt{\text{x}+1}\big)^4\\\big(\sqrt{\text{x}-1}\big)^2-{^6\text{C}}_3\big(\sqrt{\text{x}+1}\big)^3\big(\sqrt{\text{x}-1}\big)^3\\+{^6\text{C}}_4\big(\sqrt{\text{x}+1}\big)^2\big(\sqrt{\text{x}-4}\big)^4+{^6\text{C}_5\big(\sqrt{\text{x}+1}}\big)\big(\sqrt{\text{x}-1}\big)^5+{^6\text{C}}_6\big(\sqrt{\text{x}-1}\big)^6+{^6\text{C}}_0\big(\sqrt{\text{x}+1}\big)^6\\{^6\text{C}}_1\big(\sqrt{\text{x}+1}\big)^5\big(\sqrt{\text{x}-1}\big)+{^6\text{C}}_2\big(\sqrt{\text{x}+1}\big)^4\times\big(\sqrt{\text{x}-1}\big)^2-{^6\text{C}}_3\big(\sqrt{\text{x}+1}\big)^3\\\big(\sqrt{\text{x}-1}\big)63+{^6\text{C}}_4\big(\sqrt{\text{x}+1}\big)^2\big(\sqrt{\text{x}-1}\big)^4-{^6\text{C}}_5\big(\sqrt{\text{x}+1}\big)\big(\sqrt{\text{x}-1}\big){^6\text{C}}_6\big(\sqrt{\text{x}-1}\big)^6$
$=2\big[(\text{x}+1)^3+15(\text{x}+1)^2(\text{x}-1)+15(\text{x}+1)(\text{x}-1)^2+(\text{x}-1)^3\big]$
$=2\Big[\text{x}^3+1+3\text{x}+3\text{x}^2+15\text{x}^3-15\text{x}^2+15\text{x}-15+30\text{x}^2-30\text{x}\\+15\text{x}^3+15\text{x}^2+15\text{x}+15-30\text{x}^2-30\text{x}+\text{x}^3-1-3\text{x}^2+3\text{x}\Big]$
$=64\text{x}^3-48\text{x}$
$=16\text{x}(4\text{x}^2-3)$

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