Question

If $\alpha$ and $\beta$are the roots of $x^2-2x+4=0$, then the value of ${\alpha }^6+{\beta }^6$is

• A. $32$
• B. $64$
• C. $128$
• D. $256$

Answer 1

The correct option is C

$128$

Explanation for correct option:

Find the value of ${\mathit{\alpha }}^{\mathbf{6}}\mathbf{+}\mathbf{}{\mathit{\beta }}^{\mathbf{6}}$:

Given that $\alpha$ and $\beta$are the roots of $x^2-2x+4=0$.

$\Rightarrow (\alpha +\beta )=2;\alpha \beta =4$

$$\begin{gathered} {\alpha }^2+{\beta }^2={(\alpha +\beta )}^2–2\alpha \beta \\ \Rightarrow {\alpha }^2+{\beta }^2=4–8=–4 \end{gathered}$$

$$\begin{gathered} {({\alpha }^2+{\beta }^2)}^3={\alpha }^6+{\beta }^6+3{\alpha }^2{\beta }^2({\alpha }^2+{\beta }^2) \\ \Rightarrow –64={\alpha }^6+{\beta }^6+3\times 16(-4) \\ \Rightarrow {\alpha }^6+{\beta }^6=192–64 \\ =128 \end{gathered}$$

Hence, the correct option is C.