Question

The expression
$P\left(x\right)={(\sqrt{x^5-1}+x)}^7-{(\sqrt{x^5-1}-x)}^7$ is a polynomial of degree

• A. $16$
• B. $18$
• C. $20$
• D. $27$

Answer 1

Answer: (A)

$16$

Consider $\sqrt{x^5-1}=a$
Then the expression reduces to, $(a+x{)}^7-(a-x{)}^7$
$=2[7C_1{a}^6.x^1+{}^7{C}_3{a}^4.x^3+{}^7{C}_5{a}^2.x^5+{}^7{C}_{7}a.x^6]$
Now consider $7C_1{a}^6.x^1$
$=7\left(\right(\sqrt{x^5+1}{)}^6.x)$
$=7\left(\right(x^5+1{)}^3.x)$
$=7\left[\right(x^{15}+1+3x^{10}+3x^5).x]$
$=7[x^{16}+3^{11}+3x^6+x]$
Hence the degree of the above expression is $16.$