Question

The expression${\left[x+{\left(x^3-1\right)}^{\frac{1}{2}}\right]}^5+{\left[x-{(x^3-1)}^{\frac{1}{2}}\right]}^5$ is a polynomial of degree.

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C

$7$

The explanation for the correct answer.

Solve for the required degree.

$$\begin{gathered} {(a+b)}^5+{(a-b)}^5 \\ {=}^5{C}_0{a}^5{+}^5{C}_1{a}^{4}b{+}^5{C}_2{a}^3{b}^2{+}^5{C}_3{a}^2{b}^3{+}^5{C}_{4}a{b}^4{+}^5{C}_5{b}^5{+}^5{C}_0{a}^5{-}^5{C}_1{a}^{4}b{+}^5{C}_2{a}^3{b}^2{-}^5{C}_{4}a{b}^4{+}^5{C}_3{a}^2{b}^3{+}^5{C}_{4}a{b}^4{+}^5{C}_5{b}^5 \\ =2\left[a^5+10a^3{b}^2+5a{b}^4\right] \end{gathered}$$

Here, $a=x,b={\left[x^3-1\right]}^{\frac{1}{2}}$

$$\begin{gathered} \because {\left[x+{(x^3-1)}^{\frac{1}{2}}\right]}^5+{\left[x+{(x^3-1)}^{\frac{1}{2}}\right]}^5 \\ =2\left[x^5+10x^3(x^3-1)+5x{(x^3-1)}^2\right] \\ =2x^5+20x^6-20x^3+10x^7-20x^4+10x \end{gathered}$$

The given expression is a polynomial of degree $7$

Hence, option(C) is the correct answer.