Question

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

• A. 7
• B. 4
• C. 5
• D. 6

Answer 1

The correct option is A 7

Given,

$[x+\sqrt{x^3-1}{]}^5+[x-\sqrt{x^3-1}{]}^5$

Let $x=a$ and $b=\sqrt{x^3-1}$

$\therefore (a+b{)}^5+(a-b{)}^5$

Exapanding using Binomal expansion

$\left(a^5{+}^5{C}_1{a}^{4}b{+}^5{C}_2{a}^3{b}^2{+}^5{C}_3{a}^2{b}^3{+}^5{C}_{4}a{b}^4{+}^5{C}_5{b}^5\right)$

+$\left(a^5{-}^5{C}_1{a}^{4}b{+}^5{C}_2{a}^3{b}^2{-}^5{C}_3{a}^2{b}^3{+}^5{C}_{4}a{b}^4{-}^5{C}_5{b}^5\right)$

$2\left(a^5{+}^5{C}_2{a}^3{b}^2{+}^5{C}_{4}a{b}^4\right)$

=$2(x^5+10x^3(\sqrt{x^3-1}{)}^2+5x.\left(\sqrt{x^3-1}{)}^4\right)$

=$2(x^5+10x^6-10x^3+5x.(x^6+1-2x^3\left)\right)$

=$2(5x^7+10x^6+x^5-10x^4-10x^3+5x)$

Hence the degree of given Polynomial is 7