Question

Q 103 and 104 . I am not sure about the ticks . Please explain .​

Answer 1

Answer:

103) x² - 5x + 1 = 0

=>  x² + 1 = 5x

=> x + 1/x = 5

Then

( x + 1/x )² = 5² = 25  =>  x² + 1/x² + 2 = 25  => x² + 1/x² = 23

and

( x + 1/x ) ( x² + 1/x² ) = 5×23  =>  x³ + 1/x³ + x + 1/x = 5×23

   =>   x³ + 1/x³ = 5×23 - 5 = 5×22 = 110

Finally,

( x² + 1/x² ) ( x³ + 1/x³ ) = 23 × 110 = 2530

=> x⁵ + 1/x⁵ + x + 1/x = 2530

=> x⁵ + 1/x⁵ = 2530 - 5 = 2525

104)

$\displaystyle p = 2^{\frac23} + 2^{\frac13} = 2^{\frac13}\bigl(2^{\frac13}+1\bigr)\\\\\Rightarrow p^3 = 2\bigl(2^{\frac13}+1\bigr)^3 = 2\bigl(2+3\times2^{\frac23}+3\times2^{\frac13}+ 1\bigr)\\\\{}\qquad=2\bigl(3+3\times(2^{\frac23}+2^{\frac13})\bigr)\\\\{}\qquad=2(3+3p)\\\\{}\qquad=6+6p\\\\\Rightarrow p^3-6p-6=0$