Question

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

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C

$7$

Explanation for the correct option:

Simplify the given expression.

The expression ${\left\{x+{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5$ is given.

Simplify the expression as follows:

We know that, ${\left(x+a\right)}^n+{\left(x-a\right)}^n=2\left\{{}^n{C}_0{x}^n{+}^n{C}_2{x}^{n-2}{a}^2+.......\right\}$

$$\begin{gathered} {\left\{x+{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5=2\left\{{}^5{C}_0{x}^5{+}^5{C}_2{x}^{5-2}{\left[{\left(x^3-1\right)}^{\frac{1}{2}}\right]}^2{+}^5{C}_4{x}^{5-4}{\left[{\left(x^3-1\right)}^{\frac{1}{2}}\right]}^4\right\} \\ \Rightarrow {\left\{x+{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5=2\left\{x^5+10x^3\left(x^3-1\right)+5x\left[{\left(x^3-1\right)}^2\right]\right\} \\ \Rightarrow {\left\{x+{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5=2\left\{x^5+10x^3\left(x^3-1\right)+5x\left[{\left(x^3-1\right)}^2\right]\right\} \\ \Rightarrow {\left\{x+{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5=2\left\{x^5+10x^6-10x^3+5x\left[x^6+1-2x^3\right]\right\} \\ \Rightarrow {\left\{x+{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5=2\left\{5x^7+10x^6+x^5-5x^4-10x^3+5x\right\} \end{gathered}$$

Therefore, the degree of the given polynomial expression ${\left\{x+{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{\frac{1}{2}}\right\}}^5$is $7$.

Hence, option $C$ is the correct.